 I would say in his previous life, when he was more, was concerned with geophysical problems. So, and it's, again, it's something related to rotation. So the subject is the explanation of the masterman gauge spectrum, which is observed in the atmosphere and has been known for some 20 years. And it's the first time when it's derived analytically, but we shall start from the beginning. So what is it all about? In the late 70s and 80s, there was observations of kind of upper troposphere, lower stratosphere from the commercial airplanes, airplanes of opportunity. And there were two basic campaigns, one by the Global Atmospheric Sampling Program, GASP. There was a bunch of paper by Masterman Gage. And they're collaborators on the American side. The European side was presented by this measurement of Osam Airbus in-service aircraft. It's called Mosaic. The most interesting paper is by Lindbergh, where there are also many papers describing this campaign. So results from GASP were summarized in paper by Masterman Gage. And they produced something which was quite interesting. It was, they called it canonical spectra, or Masterman Gage spectrum. And this is how it looks like. It's a spectrum in the troposphere and stratosphere. And it was for zonal wind and meridional wind. And you can see that it has two branches. On the meta-scale, it goes like k to minus 3. And on synoptic scales, it goes like k to minus 5 third. So synoptic scales, I think I switched them around, synoptic scales are from several thousand kilometers to 500 kilometers. And the meta-scales are from 500 to 10 kilometers. So there was a question how we can derive the spectra. And it was very interesting to find explanation because they seem to be universal. They didn't depend on the season. On average, they didn't depend on altitude. Just universal spectra. So what wasn't noticed in this, because they didn't know how to account for the effect of Coriolis parameter. So the spectra were kind of averaged over the different latitudes, even though there were some latitudinal dependence. But it was noticed that the spectral amplitude decreases towards the equator. So this was kind of a puzzle for the years to come to solve the physical nature of the spectrum. So what was the physics or what, well, it was attempt to relate the level of knowledge with the observations. So what was assumed that the k to minus 3 branch was direct, was due to direct and stratificus k, because the macroturbulence on Earth due to large-scale baroclinic instability. So they have forcing on large scales. It can produce stratificus k to smaller scales. And it comes with a spectrum k to minus 3. And the other part of the spectrum k to minus 5.3 was supposed to be assumed to be due to inverse cascade, with some small-scale forcing. And there was paper by Doug Lilly saying that convective storms on small scales really, on the order of several hundred meters, maybe kilometer, maybe may serve as a source of energy for inverse cascade. And there was k to minus 5.3, due to this inverse cascade. But the following up simulations showed that these things didn't work. There was a paper by O'Gorman and Schneider in 2007, showing that an stratificus k was not necessary to obtain this minus 5.3 slope. Then there was a paper by Lovejoy, showing that he claims that this spectrum of this range doesn't even present. And it shows up because of the errors of the analysis. With respect to this part of the spectrum, there was a famous paper by Eric Lindberg in 1999, in which he used structure functions. And he showed that it's not an inverse cascade. It's actually a direct cascade of energy. And the energy goes from large scales to small scales. So it looks like Calmogorov turbulence. But of course, how can we understand it? It's a variant of the tropic turbulence. That's not what Calmogorov was dealing with, because the horizontal scales are much bigger than vertical scales. So it's not clear how this direct cascade can exist. So later on, Lindberg suggested that it may be stratified turbulence with a large degree of anisotropy. So he developed direct simulations with highly anisotropic grid, with much finer resolution in the verticals than in the horizontal, and showed that this may explain the spectrum. But then there was a paper by Scamarock in 2014, and he showed that this was not supported. Important paper, and it's not very often noticed, is by Choi and Lindberg in 2001, in which he said that the Coriolis force may be responsible for this. But since they dealt with the energy equation, on the level of energy equation, Coriolis force falls out of the equation, cancels out, because Coriolis force is conservative force, doesn't produce work on the flow field. And at this level, the effect of Coriolis force could not be accounted for. So the possibility was open, but there was no more research in this area. So it's still not clear understanding about this spectrum. Even if you take the very last issue of Journal of the Spiric Sciences, there is a paper by Group. I think that either Hungarian, I don't remember it, Peter Bechtel was involved in this, and they were explaining part of the spectrum by Rosby waves on large scales, and another part by internal waves. But they didn't present the spectrum itself, so there is nothing to compare us. Another problem which we have with the spectrum is that we only know the spectra, but we don't know what makes the spectra. We don't know if it depends on centrifugist heat or it depends on the Coriolis parameter or it depends on energy cascade. And it's very difficult to ascertain the physical nature of the spectrum if we only know the slope. We need to know more than just the slope. And part of this research was to find out what makes the spectrum, why it looks like it does. So in addition to this, the spectrum has more than just academic interest because it has a lot of application and atmospheric modeling. First of all, atmospheric prediction systems, they all designed and configured and validated based on the spectrum that their ability to reproduce the spectrum-engaged spectrum. Scamrock and shoots later on, they also used the spectrum for evaluating the forecast error growth, spin up time scales, filtering, and sub-finder-scale physics. It all depends on the spectrum. It depends on the physics, obviously. Moreover, the ability of the model to reproduce the canonical spectrum is very often taken as a validation of the correctness of the model formulation and the implementation and configuration and the secondary scale parameterization, all of this. Now, the kinetic energy spectrum also is a basis for developing of sub-finder-scale parameterization which is used in the models. So we can ask if there is indeed connection between model configuration and correctness of the simulated spectrum. This is one of the issues which is dealt in atmospheric modeling. And another question is, is there any connection between vertical resolution and the form and magnitude of the filtering? Now, there was more questions coming from the group of Lovejoy because they even questioned the importance of turbulence. And it's another interpretation of the spectrum. So they were saying that the failure to account for different scaling laws for turbulent processes and horizontal and vertical may lead to spurious results. And they were saying that flattening of structure from 3D to two-dimensional turbulence may account for a change in the spectrum. Now, they said that another tropic turbulence, well, jump to it a little bit more. So they emphasized the entire mainstream view atmospheric turbulence was fundamentally colored by assumption of isotropic turbulence, and which is true. And atmospheric turbulence is not isotropic. So the question is how to take it into account. It seems to be like a difficult problem. So the last in this series of paper was paper by Yanar, who just copied what he said. In summary, in spite of a peeling nature of an isotropic turbulence theory that potentially unifies atmospheric flows of all scales, as it stands it now, it remains a purely statistical theory without a counterpart dynamical model for describing the system in deterministic manner. Such a system should have a capacity continuously transforming from a quasi-geostrophy to non-hydrostatic anelasticity. And he said my naive feeling is that an elaborated use of renormalization group theory of turbulence might potentially lead to a necessary theoretical breakthrough. But he said I shouldn't be too speculative. And it was in 2010. But it was kind of a visionary remark, because this is what we did with Simeon, and it's exactly conceptually close to the renormalization group theory. There was a theory which was developed by Simeon and myself called QNSE, quasi-normal scale elimination, which yields analytical expression for, in this case, for an astronaut-engaged spectrum. And surprisingly, without any tuning, it just gave the correct result, as we will see later. So what the theory does? It considers three-dimensional fluid, which occupies infinite domain. And the dynamics is represented by full Navier-Stokes equation and continuity equation. And the coordinate system is rotating with angular velocity omega. There is no temperature here. It's just turbulence with rotation. The model of the approach of QNSE is very similar to RNG. It's basically the same ideology. It's a successive coarsening of the flow domain by cyclically eliminating small shell of wave numbers, starting at the highest wave numbers around Kalmabr of wave number. Because of this, wave numbers are effectively realness numbers of the order of 1. And then some procedure can be organized. And this small shell can be eliminated. Then the bunch of scale is removed. But the compensating effect of the scales on viscosity and diffusivity, in this case, only viscosity is computed. And then the equations are re-read with new viscosity. And the equation preserves its original way. And then the next shell of wave number can be eliminated and so on. And it can go all the way under the integral micro scale. Or it can be stopped somewhere in the middle to develop a larger dissimulation scale model. So what is important here is that its full system of Navier-Stokes and continuity equation, no quasi-geostrophies implied here at all. And we don't look on energy equation here. So the problem with a Coriolis parameter, a Coriolis form falling out of this equation doesn't bother us here. We have a paper published in JFM with the theory of some end of last year. So if you're interested, there is a first page of it and can be found there. As I said, well, just repeat, we have Navier-Stokes equation, continuity equation. I have a Coriolis term here, full non-linearity. And it's based on Fourier transform and derivation and Fourier space. So first of all, the theory produces interesting crossover wave number. You can see in the process of elimination that a scale occurs with this l omega, which is epsilon divided by f cubed to the power of 1 half, which is, we call it root scale, it's known as Zeeman scale, but roots found the scale about 20 years before Zeeman. So it's probably more to give him credit for this name. f is 2 omega as a Coriolis parameter. And at this scale, the length scale, the time scale of turbulence and inertial waves are equal. So this is a crossover scale between turbulence and waves. On smaller scales, turbulence dominates. On larger scales, inertial waves will be dominating factor. And the procedure of coarse-graining goes like this. We take some wave number lambda and take a small shell delta lambda in a way that delta lambda is much smaller than lambda. And we separate the modes to fast and slow. The fast modes are sitting here. And slow modes are sitting in the rest of the domain. And we compute the correction of the delta lambda to the inverse green function, but eventually it will become the viscosity. By ensemble averaging over the fast modes in this shell delta lambda, and the correction will generate the same order of, the procedure will generate correction of the same order to this quantities and diffusivities and diffusivities. And then it will, when delta lambda is put to zero, then we get a set of ordinary differential equations. The only thing is that in this procedure, when you have extra strains like rotation or stratification, the procedure doesn't generate only viscosity by itself, but it also can generate other viscosities which are acting in different directions. In the case of stratification, pure stratification is no rotation. It was vertical viscosity and horizontal viscosity. In case of rotation, when there is no even temperature, this is much more complicated because rotation has a specific feature to affect not only horizontal and vertical directions, but also affect the vertical velocity component in a way different from horizontal velocity component. So it's called componentalization, I think. That's invented by some French guys and cannot even pronounce them. So we have four viscosities here. Two are acting in horizontal and vertical, and two more are acting in horizontal and vertical on vertical velocity component only. And you can also add the diffusivity equation in the case, and this will give you two more equations for the normalization of diffusivity and the horizontal and the vertical. The effect of rotation is also quite complicated because it causes energy cascade to go into a large scale at some point. Some wave number, we have the quantity goes to 0, which is signaling about starting from inverse energy cascade. And at this point, we need to start the procedure. But we have all this procedure going from the forcing scale, if it's close to the wood scale, and all the way down to a small scale. As I said, this is another way to show the emergence of the wood scale. And it's coming from the spectral Rosby number, which is given here. OK, so there are equivalents of wood scale and the representations of the spectral Rosby number. And because we only take the derivation close to the wood scale, it's all done in the small parameter expansion around the wood scale. So this can be so done analytically, and there are solutions which are given here. OK, well, I'm not going to go through the solution. But what is important, there is a complete difference to the effect of rotation and the effect of stratification. In effect of when stratification acts, then you will see that the vertical viscosity and diffusivity decrease by horizontal, but horizontal, otherwise increase. But in the case of rotation, horizontal viscosity decreases and becomes zero. And then we have inverse cascade. So in many senses, the effects of rotation and stratification are opposite to each other. But it's more subtle. And I don't have time to spend on this. Let's go to the spectra. So the spectra can be derived analytically. And you can define a longitudinal and transfer of spectra. And the most interesting here are the longitudinal spectrum of horizontal velocity and the transfer of the spectrum of horizontal velocity as well. And this is more or less what Nasterman gauge plotted. So when we have this equation, you see that it has some kind of analytical coefficients. When we take this equation and plot them against the data in Nasterman gauge, this is what we get. This is the zonal spectrum, which is longitudinal spectrum. And this is the zonal spectrum of the radial velocity, which is transfer of spectrum. This is how it was actually computed. So the color line as what is coming out from the theory and all these dots are the measurements in the airplane. They're a little different because what I showed before, it was latitude averaging. This data is not average in latitude. And then we can see that it doesn't maintain as k to minus 3. In this case, it's a little less steep. But the agreement, other than that, the agreement is very nice. And this is just coming from an analytical theory without any adjustment, without anything else. The only thing which you can ask me is how do we know epsilon for the spectrum? Well, epsilon was measured by Freilich and Scharmann, at least one non-paper. There are some other measurements that are close. So this is epsilon which is used in this derivation. So it's just taken directly from the data. And as you can see, this is not inverse cascade. This is a direct cascade. And this direct case throughout the spectrum. It's assumed that the flow is forced on large scales. So it's a baroclinic instability. And the physics of this is an isotropic turbulence and dispersive waves at the combination of the two factors. The spectrum is universal. And it depends only on f, on Coriolis parameter. You can see that we don't talk about n-strophic cascade here. And we don't need it. And it can be quantified. So the first obvious conclusion from this is that this part of the spectrum should diminish and should disappear as we go to the equator when f goes to 0. This was not done by Naster-Mengage because they couldn't account for the effect of Coriolis parameter. And they didn't know anything about it. But they did give us the data. They provide some measurements of the spectrum at different latitudes. So this is their measurements. You can see the range of their data. And given in the table. And this is three lines produced by the theory for 10 degrees north, 30 degrees north, and 60 degrees north. This agreement may not be ideal here because they really didn't concentrate on precise measurements. So it's probably need to be done more accurately. But the range of variation is very, very close. And at least we understand now what by the spectrum depends on the latitude because it depends on f and not on Nsterfeer cascade. So as I said, it's really important to know not only the slope of the spectrum but also parameters which enter the spectrum. This is there even more important than the slope. Now, can we say something about the equator? There was a data by Sue and company from 2011. It's not tropospheric data is over the ocean from quick scat kind of data, say middle tropospheric and lower. But what is important here is that this is our data, global data. They separated the data on several regions. They have some north region here, southern region here, and the equatorial region is here. And they plotted the average spectra which they found in their data. This spectrum is steep on the northern part. It's under minus 3, but over minus 2 to 2.5. The same is on the southern part. But the equatorial part is very close to Kalmogorov. So as expected, the part which is Kalmogorov dependent in the equatorial region is not present here. Because Kalmogorov is very close. Because Karol's parameter is very small. Now, what we derived here is universal spectrum. It doesn't tell us that it's atmosphere. It can be applied anywhere else. This is just rotating flow and the general derivation. So why cannot it apply to oceanic flows as well? Recently, there was some interest in measuring oceanic turbulence and the oceanic spectra. And those people who did it, they distinguished between longitudinal and transferable spectra. And this is the measurements in several campaigns. And the sea clients are the result coming from the analytical theory. So you can see that even in the oceans, the agreement is very good, particularly in this case. There is one more thing. This part of the spectrum, the oceanic spectrum, is given by universal garret and monk spectrum, which is also unknown. What is the reason for the spectrum? But what is coming out from QNSE actually goes right over the garret and monk spectrum. So we may not only have explanation for this part, but we also have explanation for the garret and monk spectrum as well. It still needs to be, I was trying to say an astro-engaging, I forgot, but we may have some byproduct which is applicable to garret and monk spectrum. Now, we can apply to spherical geometry. It needs to be now rewritten in spherical coordinates. R is the radius of the Earth, and omega is an angular velocity rotation of the Earth. And there are some simulations going back to Scamero. This is what he found using the M-pass model with 3 kilometer resolution global model M-pass. So you can see that it has minus 3 part and minus 5 third in the troposphere and the stratosphere. The stratosphere is not as good an agreement, but it's still there. And there were some simulations by Peter Hamilton using his model in which he was trying to determine what is the effect of Coriolis parameter on the spectra. So what he did, he ran the general case, and then he increased and decreased rotation by a factor of 2. And as you can see, his spectra went up and down. And this is our spectra. Well, this is our spectra to begin with. He gives the spectra in different latitudes. But what can you see predicts the change of the amplitude of the spectrum by the same factor as in the case of Hamilton? They also derived the semi-scale parameterization for their model based on many empirical simulations that got some result. But if you take a Kalmogorov spectrum in this part and derive, well, they used by harmonic friction just from Kalmogorov turbulence, you will get the same dependence on the almost the same. They get minus 3.22. We get minus 3.33 dependence on the truncation wave number. So indeed, if we have theory, we can define, we can derive analytical representation for subred scales. So it's a very end. So again, we have analytical theory, which is behind the Nasterman gauge spectra. And it's done for the first time. It's based on Navier's talks and continuity equation in rotating coordinate frame with no approximations. We do use the concept on other traffic turbulence. Of course, it's very important in this case. And we account for the effect of Coriolis parameter. You can probably read it all. I don't want to repeat what I'm saying here. But this is very important. The way it's done, we don't rely on just traffic turbulence approximation. And by the way, it's done, it's possible that we over rely on this approximation. But sometimes we need to look on the full set of equation and take care of the turbulence that comes to the picture and not oversimplify the problem.