 And the topic is the passive scalar spectrum at very high Schmitt's number, especially the spectral form of at the, in the very far diffusive range. This is a joint work with my colleague, the Saito. So the, let's begin with the two non-dimensional parameters. As we know in this program, the one important non-dimensional parameter is a Reynolds number. And in addition to that, there is a one important parameter for passive scalar, Schmitt's number or Prandtl's number. The physical meaning that this is a ratio of the molecular viscosity to molecular diffusivity. This Schmitt's number, in my talk, I mostly differ the Schmitt's number. The Schmitt's number can vary quite a wide range. For the mercury, it's about 0.01, and for the salinity, it's about 700, or for the engine oil, it's quite rush. This is the example of the salinity distribution in the ocean. So you can see that the salinity in the Atlantic Ocean is quite high when compared to the Pacific. Another example of the Hirsch-Schmitt's scalar is the aerosols. And the last five years, I have been involved in a study of droplets or the growth of droplets which are converted by turbulence. So in the case of cumulus clouds in the Maritime tropical area, so the aerosol is a kind of very tiny salt blown up from the sea surface. And this aerosol come to some high altitude, about 500 meters, and this aerosol can be condensation nuclei for the water vapor to condensate on it. Without this aerosol, so the nucleation of very tiny cloud droplets is almost very difficult. So with the existence of aerosol, we can get tiny cloud droplets. After that, by condensation process, these tiny cloud droplets evolve into the size of about 30 micrometers. During this process, so the heat is released, then the upward draft is generated. And then beyond the 30 microns in the radius, so the droplets begin to collide, to form the rain drops. So anyhow, if we get at least the distribution of aerosol as a continuum, we may estimate the diffusivity of these aerosols. The typical size of the aerosol is less than about 0.01 micrometers. So by using the Einstein formula, we have this estimate for the Schmidt number, about 10,000. It's quite large. So this is the example of snapshot of the smoke aerosol from mountain fire in the Far East. So this fire, the smoke aerosol can be the nucleation of cloud condensation. Now we consider the spectra of the scalar or the aerosol at the Schmidt number. In this program, as we know, depending on the Schmidt number, there are a few scaling range for a possible scalar spectrum. And for the inertial convective range, so we know, both the kinetic energy and the scalar spectra over is k to the minus 5th. And the Schmidt number is much higher than 1 than one inertial, one scaling range exists called viscous convective range. In this range, the velocity field is already died out, very smooth everywhere. However, the scalar field can be rough because molecular diffusivity doesn't work at this scale. But after some successive squeezing or stretching, the scalar field can be confined in a very thin layer about some wave number kb, which is determined by the laser. Anyhow, the one important thing about the viscous convective range is that the time scale of velocity field is independent of wave number, which is given by just a column of time scale. Therefore, the virtual wave number is determined by balancing the viscous diffusive time scale to the column of time scale. So we have this estimate. That means, virtual wave number at which the molecular diffusivity works is proportional to the square root of Schmidt number. Now let's consider the situation of the aerosol scale in the context of Schmidt number. So at large scales, we have large turbulent eddies. After succession of cascades, the fluids are discred, become small blobs. The typical column scale in the context of the cloud is about 500 micrometers, 0.5 millimeters. This is a column scale. And since the Schmidt number is estimated 10 to the fourth, therefore, the angular scale is about 5 micrometers. In between them, raindrops plays a game. The typical size of raindrop is about 20 micrometers to 30 micrometers. For the raindrops, it's about 100 or 500 micrometers. And smaller than the virtual scale, there is some scale region in which the cloud condensation of cloud plays a game. The typical size of aerosol is about 0.1 micrometers. So there is a scale separation from here to here and here to here. So this is the scale range I'm talking about. Among them, this is the inertial convective range, but mostly I'm talking about this range, the divisive range, when the viscous convective range exists for when the Schmidt number is very high. So now I explained some representative statistical theory for passive scale in this range. The key player is that rate of strain tensor. So as I told you that, we are talking about scale much smaller than column scale. Therefore, velocity field is so smooth, therefore, we may assume that velocity gradient almost uniform over some range of scale. And then we consider the rate of strain tensor. And we take some local coordinates in which that eigenvector is parallel to the local coordinates. The eigenvector is the order in this way. The gamma is most negative, the alpha is most positive, the largest one. I say I talk the three theories. One is virtual theory. In his theory, the gamma is the most negative eigenvalue of the rate of tensor. The gamma is assumed to be constant on no fluctuation, but it has infinite time correlation. In the opposite case, in the Krugman theory, it's a famous theory for passive scale, we assume that the rate of strain tensor obeys Gaussian statistics and zero time correlation. In between them, there are Lagrangian spectral theory. This is a spectral theory based on some mathematical procedure. So it assumes that the nearly Gaussian statistics for rate of strain tensor and it has finite correlation time. Now I explain a little bit in more detail. And the bachelor theory. So in his theory, in the local coordinate, each axis is the parallel to the eigenvector. So we suppose that the velocity field is squeezing. So there is no pressure term of scalar. Therefore, scalar is easily squeezed into the small size, but at some scales, diffusivity becomes work. Therefore, the bachelor assumes that they are the balance between squeezing term and diffusive term. So he obtained this equation for the scalar spectrum. This equation is easily integrated to half this one. The point is that k to the minus one and the scalar Gaussian decay in the diffusive range. Then bachelor thought that the gamma fluctuates and therefore this gamma is replaced by some effective rate of strain rate, which is given by this one. The result is that there are some non-dimensional constants, Cb, the k to the minus one followed by Gaussian decay in the diffusive range. Now, correctional model. He assumed that S is delta correlated in time. So he obtained the equation for E theta as this. The point is that this term, second order derivatives with respect to wave number, this is the expression or this explains the fluctuation effect of straining field. And one important thing is that lambda. Lambda is a triple relaxation time, which describes the rate of transfer of the scalar from through the in a viscose convective range. Anyhow, lambda in theory, lambda is constant. lambda is Cb, k to the minus one and exponential decay. Now, spectral theory of scalar. The Lagrangian theory, like the LGDI by Krachnan or Lagrangian approximation by Kanida. This Lagrangian spectral theory contains no ad hoc parameters and the set of equation is free, obtained by free systematic way. And the equation is closed by using the Lagrangian representatives, but let's skip the detail. The result is, the result, the equation for scalar is very much similar to that of Krachnan. The difference is here. The lambda is now the function of wave number. The result is Cb and k to the minus one. These are same, but in the far divisive range, spectral decay is Gaussian way, quite rapid. This is the summary. The bachelor theory, it decays Gaussian way. Lagrangian theory predict Gaussian decay. On the other hand, Krachnan theory predict exponential decay. There is some difference in far divisive range. Now, let's look at the DNS data. This is the data by the p-case group, Diego, Sydney, and Young. So for the case of Schmid number 10, the theory is not long enough, but it is apparent that the DNS curve decays quite slowly than the bachelor. Or bachelor decays, bachelor spectral decays quite fast. And Krachnan spectra is not the same, but close to the DNS. Now, this is the measurement data. So quantum number is 7. This is the data, and this is the bachelor spectra. And the Krachnan spectra, the curve is like this one, very close to each other. So now, I come to the point to explain my motivational problem. Why is the Krachnan model so good, in respect of apparent this big difference in velocity field? As I told you, the velocity field in the Krachnan model assumed Gaussian, delta correlate in time, which is totally different to the case of actual Navier-Stokes turbulence. It's a big difference. This apparent discrepancy needs to be explained. And our current status tells us that we lack some physics to understand, explain the scholar spectra in far-diffusive range. So my position is that maybe the scholar spectra in far-diffusive range is not so important, except very peculiar cases. So, however, this apparent discrepancy between observation and the theory suggests we do not know something important for the scholar spectra dynamics of scholar spectra. What functional form is the scholar spectra in the far-diffused range at the High-Schmidt number, it's my question. This is the different statement about the three theories for the time scale and the statistics of straining field. So the virtual theory sits here for the infinite time scale. The Krachnan model sits here, the zero-time correlation, but nearly Gaussian. And the Lagrangian spectral closure tells is sitting around here. But our target is here because we are considering actual turbulence effect. Okay. To approach this problem, we did some first step, we did some computation, DNS. We integrated Navier-Stokes, the scholar equation, both are excited by random force or random injection, which has delta correlated in time and applied only low-wave number range. So the range of the parameter of the Schmidt number is 200,000. And the resolution is around here. However, because we consider the High-Schmidt number, so we need it to keep the Reynolds number quite small or moderate, like the 42. This is an example of snapshot ice contour of the scholar. So you can see that the scholar feels highly, how do you say, squeezed in small scales. However, this is a botanistic field. You can see very smooth, thick, okay? So you can see that there is clear scale separation in this problem. This is the scholar spectra and the scholar spectra. And this is normalized the scholar transfer flux, as you can see. Over the one decade of wave number, the curve stays almost one. Now this is a compensated 3D scholar spectrum by multiplying k1. So the curve stays almost horizontal line. The value is about 0.57. Now we put the curatina spectra on this curve, like this nicely collapse. Why? This is the same plot, but in the semi-organismic way, the 3D scholar spectrum. So the curatina spectra slightly decays faster than the VNS, but this is a 3D plot. But if we compute the 1D scholar spectra, the 3 curves nicely follow the straight line. And similarly to this, curatina curves like this one slightly decays faster than the VNS, but close to each other. Now the problem is that a spectral theory contains, consider the finiteness of correlation of the strain field, like this one, by introducing a divisive time scale. The only remaining factor which is not considered is the non-gaussianity of the strain field. So we consider. So remember, the bachelor theory introduced the spectrum is expressed in terms of gamma. It's the most negative eigenvalue of the rate of strength tensor. It is quite natural to compute the distribution of gamma, like this one. I'll run the 42. You can see the left tail is quite long. Now let's take an average of bachelor spectra over the distribution of DNS obtained gamma. The red curve is the DNS for the passive scalar in the far divisive range. And the blue curve is computed by this spectrum, this formula. So two curves are close to each other. And we put further the largest computation to the graph. Again we can see that green curves close to the blue one. So these correspond, the curves of the two curves suggest that input, intermittence of the strain motion is very important to have the non-gaussian, the exponential decay or at least slow decay of the scalar spectrum in the far divisive range. So far the story is about the result for moderate Reynolds number. Now we'd like to explore what is the asymptotic spectrum, scalar spectrum, a high Schmidt number and high Reynolds number. We cannot do the DNS for this case because Schmidt number is quite high, okay. So to proceed further we do the theoretical analysis with help of DNS data for the rate of strain tensor. For this purpose we use the DNS data, about 835. The, our spec, the turbulence has a spectra is like this one. So there exists some inertial range. Then this blue and green curve, the passive scalar spectra, but the Schmidt number is 0.72. But we do not use this data. We just use the data of velocity field. Now we computed the PDF of three eigenvalues and PDF, Yippie zero. So you can see the left tail of gamma is quite long. The tail of PDF, PDF tail of Yippie zero is quite long. Now for the theoretical analysis we need the functional form asymptotic tail of these PDFs. Now this is a one example. This is a both logarithmic plot for the PDF of gamma or alpha. So you can see that for small eigenvalues the PDF of this power law with exponent 4 is very clear now. And for the tail, beta, beta is obtained by the fitting the straight line in this a little bit long of low PDF. The beta is about 0.6, okay. Now this is a similar plot, but for Yippie zero. For low Yippie zero, PDF obeys power law with exponent 1.50. The tail is 0.278 or whatever. The point is that now with the help of this data we assume that asymptotic PDF of gamma like this one, a pre-factor with exponent alpha and stretched exponential. Now this functional form of PDF of gamma is the problem in this one. Now we compute this integral by using steep descent. The result is k to the minus 1 and some exponential decay with this factor. By using the value of the beta 0.6. Now this exponent is smaller than 1. Meaning that scalar spectra in the far divisible range decays stretched exponential way, slower than the exponential. This is a result by the actual numerical computation on this one. The red is by using the PDF obtained by DNS. So please note that the abscissa is with the power of 0.75. So the straight line of the red curve shows that indeed there are a scalar spectra decays and stretched exponential. I skip the detail. Now almost finishing. Now we do the similar business for the spectral theory. Because we are considering the scales are far below the virtual scale, which is far below the columnar scale, therefore scale separation. Then we consider the equation which is obtained by the spectral pleasure as the equation conditioned by epsilon. So by solving this equation, we have some dissipation spectra in far divisible range which is Gaussian decay. Then we take ensemble average over distribution epsilon. The result is again we have this factor. This exponent is smaller than 1.7, very close to the previous one. So this suggests that when the Reynolds number is very high in the far divisible range, the scalar spectra decays slower than exponential. And if these stories are true, then this suggests that even the scalar spectra at a second order moment is not universal, strongly, weakly dependent Reynolds number. Let me summarize. The E theta is affected by the intermittency of the strain motion and the functional form is characteristic exponential. And the power is weakly dependent on Reynolds number. And the spectral tail obtained by DNS and exponential experiment data are not long enough in the divisible range. The collage in a spectrum is happened to be closer to the DNS, my understanding. And probably Batchelac constant is also dependent weakly on Reynolds number. So I have to click. Thank you very much.