 This kind of concept, semi-ragurantism concept, studied not from 1980s. At the time, I involved in this kind of dynamics. But it studied from 1970s. Yes, it was studied from 1970s. I think in 1980s, early 1980s was a very exciting period. And if you read the paper in early 1980s, you may find very, very interesting papers. OK, anyway, this is the conservation. One conservation. And here, through the break, many people asked me why V bar equal to zero. The reason is that I said, we have a submission, OK? Excuse me, sir. We can't hear you in the recording. Whoa! Sorry. Because you're in the room. It's not a problem. But in the recording, we can't hear you. Oh, OK. OK. I mean, this week, we had tape recording of the lecture, so that I have to use this. So the assumption that I use this non-divisionalistic equation from hydrostatic, which comes from length scale, is much bigger than depth scale, which is OK. And the density constant is all kind of OK. And then, rigid top, rigid top, height doesn't change. So that height doesn't change means that, horizontally, non-divisional, this non-divisional concept. And this one gives u equal to minus d psi dy, and v equal to d psi dx. So that, if you take John Armin, v bar equal to 0. So that here, this is because of assumption. It is not always 0. Only 0 for this system. This non-divisionalistic equation case. And this equation, actually, this equation made predictions using this equation. Actually, basically, this equation, a little different. But basically, this equation. And they made good predictions, a few-day predictions with any computer. That is starting point of weather predictions. So this equation is not bad to use. People think, oh, this is most simplified equations. So that, I mean, this is theory. And the observation is different. No, not at all. I mean, these simplified equations can be applicable to many, many real nature. And so now, as I said, du bar dt over du bar is balanced by this, which is v prime zeta prime bar. So this is vorticity order of action. So that, for example, I mean, along the circuit, vorticity order of action over one circle, v prime zeta prime bar. Vorticity order of action. But this is a limited domain. Means that this is stock theorem. If you apply stock theorem, vorticity order of action changes the circulation over the boundary. Because the vorticity over here is canceled out so that only circulation is changed. So that this is circulation theorem, which comes from stock theory. So vorticity order of action changes the circulation around this circle. So this explains the circulation theorem. Now, let me, what is, what is, OK? And any mean flow interaction I talk about. And then I will talk about a very important concept of the wave activity. OK? Now, if you write the equation, this one, dt zeta prime square bar equal to plus, or plus, plus dy d minus dy dA. Minus dy at u prime v prime bar, OK? Equal to 0. So we know that you are using this u prime v prime bar in the wave flux vector. And in aliasing palm flux, y component of aliasing palm flux is this one, u prime v prime bar, negative. Now, let's drive. Because in this system, we don't have horizontal motion field, vertical motion field. So that only a meridional component is this one. Meridional component wave activity. Where is this? This erase. So let's define this as m, should of momentum. So that this is dm plus we just define this flux, OK? Plus dy dF equal to 0. Now, we can express m, should of momentum, or wave activity conservation. If f equal to gy, OK? So that if we prove this one, you can substitute this one. Then dm plus gy dy dm equal to 0, which is dT dN conserved with group velocity movement. If m moves with group velocity, which is m, m is conserved. So we have to prove this. If we prove, then flux, this u prime v prime bar, this one, flux vector with respect to group velocity. So that means that this u prime v prime bar is the direction of wave energy flow, group velocity. So then wave activity is conserved. You follow this? Anyway, I think I wrote here, there, there here. So you can read this. And then I drive this one, u prime v prime bar, u prime v prime bar minus. What is this? I wrote sin function equal to sin function explained by kx plus ly. As I said, it has real value, right? So sin function is this one. So u equal to minus dy d sin function. So real psi is 0. dy d means minus ir plus ir exponential i kx plus ly. Now, what is real a? Means a plus a conjugate, right? Real a is 2 over a plus a conjugate. A conjugate means that minus imaginary part becomes just gives minus sin. So 102 equal to ir psi x plus ly plus minus here. So that minus ir exponential minus i kx plus ly. So that you have to know the notation without this. You never drive this equation. So you have to know how you drive this equation. So in the literature, you have to drive equations. In order to drive equations, you have to know the notation. I mean, you never see the equations like this. But you have to know the real of complex value is this one. And you have to state these equations like this. And then apply to these equations. This is what you can see after this here. v prime equals this one. And v prime is this one, as I said. And v prime bar, if you multiply these two and taking journal mean, wave components journal mean, for example, exponential i 2 kx plus ly. This is still wave component, means equal to 0. So this component should be canceled out. So only case is i component multiplied by exponential minus i component can be canceled. So only this combination multiply this one and this one gives real components. And this one and this one gives real components. And finally, you obtain this one. v prime bar equal to minus even k l scale here scale. So as I said, l is always positive. This is our definition. x component should be positive always. This is very important, anyway. So I obtain this. This minus is true. Oh, yeah, this u prime v prime is minus. So it means that u prime v prime, u prime v prime bar, momentum, momentum of the direction of u prime, proportional to minus f. It's positive. All this is number, positive. And k is positive so that it's proportional to minus. So this means that momentum flux, momentum flux, no sort of momentum flux, means that we're propagating southward. So wave is inclined like this. Inclined like this means that wave is propagating this direction. Wave propagating this direction meaning that the momentum flux is no sort. I can easily compare to u prime v prime bar here. u prime should be bigger. And v prime is, because comparing this one and this one, u prime bar is negative. Here is east, here is west. So with respect to zonal mean, u prime equal to positive. v prime is positive. But u prime and v prime is also negative, southward. So that u prime v prime here is positive. And here u prime v prime equal to positive. And you add them all together, because of zonal mean. Then it's positive. So that wave inclined like this, then momentum flux, no sort. And wave can incline like this, then wave is propagating this direction, means that momentum flux is like this. So this is very important, which is momentum flux and wave propagation has opposite direction. So if the kid has a question, maybe you can do this later. I don't know. Sure. OK. If you look at the observed momentum, any momentum flux, it's mainly northward. Mainly northward. Which means the waves are tilted in this way. That's right. Is there, I mean, can theory, this theory explain why this must be the case? Must be the case, yes. The reason is that we have stationary wave through Himalayan mountain. Himalayan mountain around 35 degrees, 40 degrees. And which creates the wave propagation for selection. Stationary wave can be generated south and northward. So that momentum, this pattern can generate a wave like this. And even in subtropics and exotropics, baroconic wave, wave is like this, right? Means that the wave propagation is like this. So that momentum propagates like this. This case, a baroconic wave, is developing case. And if you consider some occult case like this. So that it's open occult in Aleutian law when the baroconic wave develops in the virtual phase. And then decay, the baroconic wave is like this. Means that the wave is propagating this direction. And the wave propagating this direction, right? Because of this, this is a wave line. So that momentum is convergence like this. So that journal mean is accelerating. Means that addy activity is decreasing. But journal momentum becomes larger. Means that this occult pattern gives journal momentum acceleration given by wave activity. So that this is decaying phase. And because of Himalaya Mountain, a wave propagating to the equator, which creates the southward propagation, which makes angular momentum flux northward. So that it takes the angular momentum from tropics to subtropics and exotropics. So this is a very important concept. And very interesting that Himalaya Mountain locate in 35 degrees. So that angular momentum can be pumped out by waves from tropics to exotropics, OK? So this one is pretty simple, very simple. But it indicates many things. And then, as I said, we have to prove this equation, OK? This one, F equal to g y g y n. And I obtained the flux F equal to minus 1 over 2 k L over chi-square square. And by applying this Boutisti equations, and Boutisti square is like this, you can derive this equation. And g y m equal to minus, equal to half of k L size square. So that the same, OK? So that minus u prime v prime bar. So which is this one. So I proved that dt dm, automatically by group velocity, equal to 0, OK? Then flux F can be minus u prime v prime. So that flux, u prime v prime bar is parallel to group velocity. So that this u prime v prime bar is very important to a diagnostic variable, OK? To a diagnosis of the wave propagations and momentum conversions, wave meaningful interactions, everything, OK? This component is important. And I think many lectures followed my lecture talking about this component very much, and also dyd. So we have to remember this property. Now, this is what I talk about. Before this, I like to talk about barotropic instability. As I said, dt d equal to 0. This is what I talk about. So that this is conserved. This is conserved. Instability means, instability means, boticity, OK? Boticity prime square can have infinity number, OK? And u bar is a limited value. But instability, OK? When instability occurs, boticity becomes infinity, OK? But I mean, it should be conserved. This value should be constant, constant for certain domain. In order to be constant, this one, this infinity value should be canceled out by the other infinity number. So that boticity should be canceled out. Infinity number has to be canceled out. Means that gamma has to change the sign, OK? So that positive infinity is canceled out by negative infinity. So this is the instability condition. So gamma changed the sign. Here, beta changed the sign. Gamma changed the sign is the necessary condition for barotropic instability, OK? So actually, this is cool Rayleigh instability. They also drive this equation, but in a very complex manner. I think when I was a student, I read Cooler's paper, which was very difficult to understand. But here, I draw this equation in a simple manner. So this is concept of instability. So gamma is beta minus d y squared d squared u bar, OK? And since if u bar is not constant, gamma can change the sign in a certain domain. So the integration of this over the domain has to have a constant value to conserve so that gamma has to change the sign. Infinity value should be canceled out, OK? OK, now this is so far I talk about the very basic property of loss wave. Now I'm going to talk about the very indexed and the loss wave, two dimensional loss wave. Still use the same equation plus u bar. Better? No, here u bar equals 0. Still use the same equation, OK? Now this is the equations that, let's see, use better, that roots be used, OK, a long time ago. Now, and he obtained a general solution. Now what about if y has changed in sign with respect to y, OK? Actually, in the y structure, I mean, journal mean structure is like this. This is u bar, and this is y. And the jet stream and in the subtropics near the equator, u bar is negative value, trade wind. So u bar constant is not good approximation. But locally, over here, it can be constant. But over here, there is a large change. If you can see the planetary scale waves, which has several tens of degrees of latitude, u bar cannot be constant. So the theory about meridional dispersion, two dimensional loss wave, so-called Ray theory, was driven by Hoskins and Carroll, 1981. Jess, I was very much amazed with this paper. At the time, I was student, graduate student. And actually, this is Carroll's PhD thesis. And now, let's consider the science, development of science. Without this, let's be, I mean, draw the solution. In 1938, let's be, 1938, OK? And then, maybe about 40-some years later, they lacks this constant, u bar constant to u bar as a function of y. And then, he draw the solution using WKP approximation solution. So that science progression is like this, 40 years. Everybody think, oh, this is simple. But it took more than 40 years to drive Ray theory for teleconnection theory. I think you can also easily can be famous. You can easily think, oh, I can accept u bar constant. And then, you like to drive in many other equations. And you can be famous. OK, now, let's talk about this two dimensional loss wave. His two dimensional loss wave, Ray theory. And based on the vocator projection, the equation is this one, equation one. It's same as this equation. Exactly same as this equation. And this is spherical coordinate. The beta m is defined by this. And u m defined by u bar divided by cosine phi. And the bonus equation is exactly same as this one. But u bar is function of y. So that the dispersion relationship is same as before. And as I said, dy with gx, gy, as I said, k over l means that the slope, wave propagation slope, the slope, this is the x direction propagation and y direction propagation, gy, gx. This is setter. This is setter. So that for the short wave, for k is large. Large number of wave number means that short wave. And the planetary scale wave, k is small. k is small means that setter is large, so that wave is propagating like this, so that the setter is large. That's why k, planetary scale wave, can propagate to the north. But synoptic scale waves propagating mostly journal propagation. As you can see there, small scale waves is propagating like this, more like a journal direction. But this is monthly mean chart. It is more like a medial propagation for the planetary scale waves because of this. Now, for stationary wave, stationary waves, as I said, stationary wave, we are talking about mostly stationary wave. k scale plus l scale equal to u bar over beta. This component is important. As I said, stationary wave combination, wave selection is k and l should satisfy this u bar over beta. We call this a stationary wave number, stationary wave number, which is determined completely by journal mean. It's journal mean of wind. Beta is also affected by here beta n is beta, actually, minus dy squared dy, d squared d maybe m. So beta m also affected by mean flow, and actually wave selection only at this. So l scale is medial propagation. That's k scale minus k scale. So l scale means l. If l is imaginary value, then it dams out, right? Exponential i ly. If l is imaginary value, it can be damped or it can be amplified, but boundary condition eliminate amplification, so that always damped out. So l square should be positive for wave propagation, medial propagation. It should be positive. So this stationary wave number is the constraint for the medial propagation. Only wave less than this stationary wave number can propagate to the medial direction. So it means that planetary scale wave, k is smaller, small value, so that medial propagation can be possible. But small scale wave, which cannot propagate, because k square is larger than k square, so that l square is negative. So this is very important concept. This is stationary wave concept. And when l square equal to 0, which means that k square equal to k, this is turning point. l equal to 0 means that there is no medial propagation, only journal propagation possible, so that wave is propagating like this and then turning. This is the latitude when ks equal to k. So this is turning latitude, turning latitude, which is very important, turning latitude. Now I plot it here, this one. If you apply EOF analysis of journal mean, you easily obtain the pattern like this, this one. And then you select positive months, coefficients positive and coefficients negative. And then making composite, then you can easily obtain this state, journal mean state like this, and journal mean state like this. Jet stream is the state of stronger jet stream case, and this is broad jet stream case. So actually this is obtained from first EOF mode of journal mean, very, very like this. So that one state to the other, the change is substantial. This is the first principle mode. And then if you compute ks, ks, stationary wave number, is like this. This is very important. Stationary wave number is almost monotonically decreasing. So that in the high latitude, ks equal to very small number. ks is small number. But in the subtropics, ks is relatively large. So that when a wave excited in the subtropics, this wave can maybe wave number 4, it can propagate all the way to here. And then it cannot propagate and turn. And wave number 2, it can propagate all the way to here. But for these two states, this wave can propagate all the way to here. But in this state, it can only propagate up to here. So that wave propagation, I mean, the turning latitude is different. And then let's derive this equation, the solution of this equation. Now, previously, we solved this equation, exponential i kx plus ny minus 1 by t. But now, u bar is function of y, so that a general solution cannot be a Fourier transform for y component, so that we have to eliminate this component and its function of y. So that you substitute this general solution to here. And with some manipulation, you obtain. So this is a general solution, as I said. And then with some manipulation, we obtain finally these equations. And with wkp approximation after this, wkp approximation, we introduce y equal to epsilon y, which is a stretching coordinate. For example, this is the medial structure of u bar, which has some strong medial displacement. But with the coordinate stretching, it is something like a small, slow variation if we stretch it 10 times. So that we can think that each locally can be constant, so that it gives some approximation. I hope you read the wkp theory. And it is straightforward. And the first approximation solution is like this. And you can substitute this. And then I draw all the way. So you can follow all this. And then final solution. I will discuss final solution here, this one. So that final solution after this, all the solution psi equal to proportional to maybe constant root exponential i kx plus Ly minus omega t. So this indicates a phase. And this is amplitude. Amplitude. But amplitude here is very important. It's function of l. l means l is ks u bar over beta minus k square root. So if k is, as I said, this is your wave number that I show here, ks equal to kk, this wave number, l is 0. So that in the turning point, l is 0, means that wave amplitude is infinity. This is important. So away from the forcing region, the wave amplitude is termed by basic state. So here, I published this one. This is very straight for the few-page paper in just 1990, January 15th just paper. I'll talk about this theory here. And the turning point is so important. And this one, same equation, but with different basic state. One is when I adopt this basic state and I adopt this basic state. And then I try to simulate the wave wave, barotropic wave solution with forcing over here, over here. And this is the solution. So that by forcing barotropic-potistic forcing over here, wave is generally like this. So that wave is propagating like this. And in the turning latitude over here, the reason is that the turning latitude is here. In this case, the turning latitude is close to here. And here, turning latitude is all the way to here. So turning latitude is different. And so here, the maximum amplitude is over here. Forcing location is over here, but maximum amplitude located over here. Because of turning latitude is located here. Usually, we can expect that wave is damping, decay, away from the forcing region. But amplitude is large over here. And the other case here, the wave is propagating like this and turning here. And turning latitude is located over here. So that maximum amplitude over here. So that maximum amplitude, this one, and this one is different. The pattern is different. So that exactly the same, all the forcing same and same set of equations, but only difference is basic state, u bar structure difference, which is observed in the observation. So that stationary wave can be changed with respect to different basic state, like this, substantial change. This is what I talk about in this paper. So for example, PNA, PNA. PNA is excited from the divergence in the tropics, tropics and subtropics. In the bodice equation minus fd, this is forcing. So that in the equator, actually forcing is almost zero, because f equal to zero. So that subtropics, little of the equator, the divergence, excite the bodice state. Forcing location is just of the equator for the linear case. But large amplitude occurs in North America, in the Pacific, in the latitude, about 40 to 50 degrees than those. This is because turning latitude locate there. So that also Atlantic mode, wave propagation, maximum amplitude located in high latitude is because ks is small value there. So you understand why large amplitude occurs remote from the forcing region, because of air. So this is the linear propagation concept. What's 30 minutes more? Coffee break? Maybe can I talk just five minutes, and then I will talk about multiple equilibrium in the next five minutes more to derive equation. Now, I talk about the freeway so far. Now, we are talking about multiple equilibrium, which is the carbon equations. Multiple equilibrium is this one, this equation. So that here, the main equations is changed by free bodice flux. And then this is kind of the forcing by mountain. And then damping. And I'm going to talk about this two wave component, how these two terms influence the basic state, and how steady state can be choose by this, particularly this component. And to understand this, we have to understand the force wave, mainly by topography. So I'll talk about in the next one hour, I'll talk about topography, force wave, and then multiple equilibrium. OK, let's have a photo session. Thank you very much.