 Namely, there is outer core and atmospheres of giant planets. They're inside of the atmospheres of Jupiter and Saturn. What happens there is that the interior of the planets is very hot. So the atmospheres are heated from the inside. And they're also cooled from outside by a radiative cooling because the heat is just radiated into a cosmos. And so quite suddenly, there is a convection going on there. So-called deep convection because the layer, the atmospheres are quite thick. So far, our understanding of the deep convection mainly comes from the numerical models. There were some earlier lab experiments and some theories which kind of predicted the behavior of those flows. But nowadays, we mostly look at kind of general circulation type numerical simulations of the atmospheres. And there is core as well. So there's nothing wrong, of course, with numerical simulations except that they're quite dissipative, in fact. So if you take the eckman-Nam, a typical eckman-Nam of the simulations, it's about 10 to the minus 6. So eckman-Nam is the ratio of viscosity to Kerourez force, 10 to the minus 6. And this is just kind of the newest, the latest, numerical simulations, 10 to the minus 6. Before that, the eckman-Nam would typically 10 to the minus 4, 10 to the minus 5. So the dissipation is quite important in those models. So the researchers, they need to overforce their models. So the heating fluxes, there must be many orders of magnitude higher than there really are in the planus. To be able to get the velocities, which are comparable to the real velocities in the atmosphere. By the way, the eckman-Nam in, for example, in the Earth core is on the order of 10 to the minus 15. So there is still nine orders of magnitude to go. So we can talk about the validity of the numerical, the results of numerical simulations only hoping that in the asymptotic case, they give us the right picture. So there are two types of general numerical modeling of planet Earth atmospheres. There are shallow water type versus deep convection kind of dynamics. Shower water models usually are used to generate zonal jets, just like those bands on Jupiter or Saturn from mainly two-dimensional like turbulence. But they have difficulty to give the equatorial jet, for example, on Jupiter or Saturn. Those equatorial jets, they are prograde. So they rotate in the same sense as the planet. And so they rotate faster than the planet. Or you can call them cyclonic as well. Deep convection models, they don't have any problems with reproducing the equatorial jets. So the question we ask ourselves is, can we offer some kind of general theory which would just add to understanding of what's going on. There's some basic physics. And perhaps in the future, contribute to some kind of parameterization of this small-scale convection in the atmosphere so that it can be used in those general circulation models. Because we can't really hope that we can resolve all those fine scales in numerical models anytime soon. So this talk is mainly about theory. But before I'll talk a little bit about numerical, sorry, laboratory experiment. And we needed to do this experiment just to convince ourselves that one of the key elements of the theory actually works. When I start talking about lab experiments in oceanographic meetings, I sometimes feel like a dinosaur. Because not many people do this rotating tank experiments anymore. But anyway, I'll try to convince so that in this case it can be useful and even be superior to numerical models. So we have a rotating tank here. It's 1.1 meter in diameter. And it rotates very fast. The period of rotation is 2.6 seconds. It's just a big thing. And it rotates very, very fast. So the surface of fluid, of course, is parabolic. And we insulate, thermally insulate the tank, except at the top. So the top is just open to the atmosphere. We can heat water by heater at the bottom. And also when we heat water, it starts cooling from the top, evaporates from the top so it cools from the top. So mainly what we did is we heated water before the experiment. And then during the experiment, we just observed the effect of cooling from the surface of this warm water. Warm water has an additional advantage that its viscosity is very low. It's about three times lower than the viscosity of water at room temperature. So we can actually reduce segment number to very low values. It's about 10 to the minus 6. So in this sense, we are doing better than numerical simulations. And also, well, if you would like to do this kind of simulation, it might be challenging because it was three-dimensional flow. And the scales of the flow are very small. So when cooling happens, the very small, very small dense parcels at the surface appear and sink. And they create a tiny vortices. So if you look at this picture, you can see those dots. And they're about 5 millimeter in radius or diameter. So we also have a thermal camera here. And you can see the temperature field in the tank as well. So we use a new method of measurements in our lab experiment. Because in lab experiments, it's usually a problem to measure velocity field with reasonable resolution. So we use the altimetry just like the altimeter used by oceanographers, the satellite altimetry. So we measure the elevation of the surface. So the elevation eta. So any flow within the tank creates pressure field. Pressure field creates the surface elevation. So the certain variation of the elevation is called eta. Here, and this is what we measure. We use optical method. So it's decoded by color, the gradient of the elevation. So the parabolic surface of the tank is basically used like a telescope mirror to amplify those tiny perturbations. The perturbations are very small. They're 100 microns. So you can't really see them by naked eye. But using this method, you can. K. So we measure basically pressure field, the gradient of pressure, because the surface elevation converts into pressure. So what can we do about velocity? So this is rotating fluid. So if we just start with the equation of shallow water to the equation. In the first approximation, of course, we can obtain the geostrophic velocity. So we ignore the time tendency and the non-linear term, because the Rosman number is quite small. And we get the geostrophic velocity. Geostrophic velocity is just a combination of the gradient of the surface elevation. So it's direct measurement of geostrophic velocity, if you like. And then we can do another approximation by plugging in this into the original equation. And so we get two additional terms here, which comes from the time tendency and the non-linear terms. So it's a geostrophic approximation. So we measure the pressure gradient and we can obtain velocity in a geostrophic approximation in our experiments. So the resolution is quite high. We resolve just half a millimeter resolution. And we get lots of data, hundreds of gigabytes, typically. So in this sense, it also can be compared to numerical simulations. So this is our analog computer, if you like. We also used, to our surprise, this altimeter method didn't work for some specific aspect of our experiment. So we needed to use something else to get real velocity, not quite geostrophic, real velocity. And we use PIV. So you're probably familiar with PIV, it's particle image velocity. You just put lots of tiny particles in the water and you follow the motion. How passive? Well, they are small and the flow is relatively slow. So I would say they are quite good in this sense. There are lots of literature on PIV which kind of confirms that. And in our case, we have reasonably sure about that. So the poly-italian particles, they float just above the surface of water, actually. So this is a typical movie from the experiment. It's not much to see, but if you look closer, you can see those tiny particles at the surface. And you can see variation of color. And you can see what's going on, some small vortices. And if you look even closer, you can probably see the zonal jets which develop there too. And there is another movie. This is from thermal camera. This one doesn't play. But anyway, it's the same thing. So you can see all those variations of temperature. They are tiny, tiny, tiny tornadoes, tiny vortices. So this is what we see. This is the total field relative to vorticity here. It's normalized by planetary vortices. So this is basically a Rosbin number here. You can see it is below 1. And the average Rosbin number is about 0.1, 0.2 perhaps. So if you zoom in into those vortices, so this is what you see, those typical vortices on the order of 1 centimeter and below size, we can resolve the velocity, et cetera. And this is the azimuthal velocity. You can see kind of like zonal jets. They are not well formed, like those jets on Jupiter and Saturn, because we are not quite there in the regime for those jets to form. But they are there. So for comparison, you can see the polar regions of Jupiter. This is a very recent picture, which was obtained by these flybys of the spacecraft above Jupiter and Saturn. What you can see here is again those tiny vortices. They are everywhere there. On Jupiter, they are a bit bigger. On Saturn, they are very tiny. Also on Saturn, there are other interesting features like very intense polar vortex in this structure, too. But I am not going to talk about them here. So this is what happened in the laboratory tank. We have those perturbations of cold elements which sink. And when they sink, so they sink along effective gravity. Effective gravity in the lab is regular gravity, which was centrifugal force omega squared r here. So it's not aligned with the rotation axis. So rotation axis is vertical. Effective gravity is not. It varies with radius. So the parcels sink not vertically, but at an angle. But the Taylor columns, they generate. They generate Taylor columns by emitting inertial waves. They are, of course, vertical, or rather aligned with the rotation axis. So this is the key feature of the experiment and the theory as well, so that the gravity is not aligned with rotation. On the planet, it's just regular gravity. Well, effective gravity centrifugal forces have included that, too. And of course, it's not aligned with rotation as well. But in a different sense. In the lab, it goes like this. And on the planet, it goes in the opposite sense. But it doesn't matter. OK, so when the parcels move in the water, it turns out they move with constant speed, with constant velocity. Because they emit inertial waves, so they experience drug. There were some previous papers, starting from paper by Steuerson in the 50s, where people studied the motion of objects in the rotating fluid, objects moving along the rotation and across it. And they showed that there is a drag force experienced by those objects, either solid objects or fluid or liquid objects, doesn't matter. And eventually, those objects, they move at constant speed, which means that the buoyancy force experienced by those parcels, by those blobs, they are transmitted to the bulk fluid by the mechanism of inertial waves. So this is a completely inviscid mechanism. So it doesn't depend on the equipment number. And it results in this forcing of the bulk fluid. So and then we can write down two equations for the general circulation in the tank, which are just regular equations of motion with the cadaverous force here, a little bit of dissipation kind of relief friction here, and this extra forcing in the right-hand side. So we neglect it. Not in reality here, it's human that the Rosman arm is small. And this gives the circulation like this. So this is azimuthal velocity u. It's proportional to this extra forcing. And there are also some radial velocity, but it's quite small because this relaxation time is quite large. Another effect which those parcels moving in the water generate, they distribute some temperature field. They move it around. So there might be some variations of temperature in the tank. And then we can calculate thermal wind, geostrophic thermal. So this is the geostrophic part of the flow. This is a geostrophic part of the flow here, which is given by the external force. So the total flow is the sum of those two. I think I need to accelerate. So you can compare this with the Ekman theory in the ocean. So when wind blows above the surface of the ocean, it transmits momentum to the water. And then this flow spirals down, so famous Ekman spiral. But the end result is very simple. The transport, the integrated velocity over the depths, is actually to the right of the wind and proportional to the wind stress. So the same exactly the same as here. But the force is coming not from the surface of water, but from within due to those inertial waves. And this is what we measure in the experiment. So we can measure the thermal wind using our thermal camera. So thermal wind is not very large here. It just goes around zero there. But we also measure this circulation here. This is a velocity function of radius of the tank. So it's approximately linear. And we can compare it with theory. What theory is predicting? So theory, we need to know this parameter, which is the product of the buoyancy force by F. F is the concentration of those buoyant parcels in the water. So we don't really know it. But we can estimate it by measuring the delta T, temperature, RMS temperature variation in the tank. Again, using the thermal camera. And we put some coefficient C here, order of unity coefficient here. And it turns out to be from all those measurements, it turns out to be 0.75 in the experiment. So now we're reasonably sure that this force actually works. And it drives water in the laboratory tank. So this force is due to misalignment of gravity and rotation. So let's now look at the spherical planar. So let's say we release particles from the inner surface, or so buoyant particles from the inner surface, or heavy particles from the outer surface here. And let's see how they move. So we have the results of all those theories there, which gives us the nice analytical expressions for drug force experienced by those parcels. So there is drug. It's different for parcels moving along the rotation axis and across. There is corollaries force. And there is also lift force, which acts sideways on the parcels. So if we put them all together and invert this expression here, we can calculate the velocities of the parcels. So we get this nice expression. So we get the radial velocity. So this is cylindrical coordinate system, by the way. So the radius here is z-axis like that. And while theta is collated here, we will be using it too. Anyway, so we get these expressions for the velocity of the particles. So we will be using vr and vz, mainly, just in the medidonal plane of the plane. We don't care about how parcels move in the azimuthal direction. So the interesting feature of those velocities of vr is that they are different by a factor of 2, exactly. So this theory, of course, is without magnetic field. But there is also a similar theory by Moffat and Loper. They include magnetic field into consideration and did similar analysis towards Sturzen did. And they obtained the coefficient for the case of magnetic field, for the teroidal magnetic field, actually. And it looks quite similar to here. And also, the same thing, it's actually a bit mysterious to me. But maybe there is some simple explanation to this. So the velocities are still different by a factor of 2. So it works for cases without magnetic field and with magnetic field. So now we can easily write down the equation of motion for particles, dr dt, dz dt equal to velocity, which is given analytically. Velocity depends on the gravity. Gravity varies like g sine theta. Theta is the co-latitude, the components of gravity. The r is that component of gravity. So you can easily see that the trajectories of particles are parabolas. So they start from inner sphere or outer sphere and they go along parabolas like this. And we can obtain this coefficient from initial conditions depending on where you release the particle on the inner sphere or outer sphere. So now we know how particles move within the atmosphere. So when particle moves, they carry the buoyancy with them and they also impart this force on the bulk fluid by emitting inertial waves. So now we can see what effects of those particles, particle motion is there. So we can easily write the flux of the entropy here. So we use entropy here for the atmosphere because entropy of the adiabatically rised particle is conserved. Or we can, in the Boussines clue, it can be just delta rho, it doesn't matter. So we can write it like this and express in the form of buoyancy. So this G-A-I-G-A expresses how particles focus into. So if we release particles uniformly from the inner sphere, for example, they rise like this. So they focus towards the pole of the planet. So this is the focusing factor. So they carry, for example, these are warmer particles. So they carry warmer temperatures towards the pole. So all these values can be calculated. Well, these are transcendental equations. So we can't really give them analytical formulas, but they're very easily calculated using some simple MATLAB or Mathematica software. So we can plug it in and calculate the exact trajectories of the particles and kind of bulk average temperature of the fluid. So the only thing, the parameters that we need to know is the buoyancy with which the particles start. So how do we guess that? We can look at, for example, thermal images. This is a thermal image of Saturn, the polar image. So again, we can, again, use this idea about the RMS temperature variation here. So the typical variation of temperature is about one degree divided by T is about 100 degrees Kelvin at the top of the atmosphere. So you can get some rough idea about the buoyancy of those of those parcels. They're kind of distributed buoyancies. So this is the factor F here, again, which is the concentration of those parcels. We can't really say anything about what's happening in the deep interior, though. So we just guess that perhaps it's proportional to this. So we introduce these coefficients, A1 and AI and A0, just to three minutes, OK. And OK, so these are the same formulas. So again, we have thermal wind and this circulation due to forcing. And this is what we get. This is for Jupiter. This is the total circulation of the atmosphere. And this is the top of the atmosphere where we can compare our profile predicted by theory with the measured profile. So the red line is our theory. And measured profile is the black line. So it's quite good. Well, we use a little bit of adjustment because we don't really know about the inner sphere. But still, it works quite well. And the same for Saturn. So it gives nicely the equatorial jet. And we also can do similar thing with the Earth's outer core. Earth's outer core is very important for dynamo problems, of course, for the generation of magnetic field of the Earth. And it works in the same manner, although the boundary conditions are a bit different. So I'll skip that. And so this is the profile it predicts. It has two anti-cyclonic jets at the tangent cylinder here. And it looks like this kind of behavior is actually observed in the data on the variation of the magnetic field of the Earth. This kind of strong jet going in the anti-cyclotic direction in the Earth's outer core. So I'll just put these conclusions here. Thank you.