 Thank you, Ricardo. I'm going to talk about fundamentals of atmospheric dynamics, the fundamentals, how I can cover with three-hour lectures. So I have to focus on something. So today, I'm going to talk about Beijing knowledge behind the multiplicative review. Where is the pointer? Ricardo, how I can go? How I can use this? Ricardo, how I can use? I want to project? Oh, OK, thank you. So fundamental knowledge behind the multiplicative equilibrium. Multiplicative equilibrium comes from a basic state, an interactive with the force waves. So I also introduce some basic concepts of free waves so that the roast wave dispersion, wave selection, and some baroconic instability, and very general dispersion and teleconnection dynamics. And then a force-roast wave, and then a multiplicative equilibrium. So I'm going to talk about basic wave property, and then a force-roast waves, and then how multiplicative equilibrium comes from this wave mean from interaction. Mainly, I will talk about the first multiplicative equilibrium theory proposed by Chani and DuVall in 1979. So I will cover that paper. But I will handle a very simple equation, most simplest possible equation, to deal with the roast waves. But I'd like to introduce how we can simplify the system, how we can obtain the simplest possible equation from full geophysical fluid dynamics equation. So from systematic approximation of Galvin equation, I obtain a simplest possible equation, which contains all the roast waves, and then I follow. Before my lecture, I'd like to mention just one thing. My lecture is a kind of classroom teaching lecture, nothing to do with the state of art research results, but introduce basic knowledge. The reason that I prepare the lecture like this is from my past experience as co-director of several summer schools at ICTP. We usually invite world top scientists as lecturers here. And all lectures are wonderful. Like yesterday's lecture by John, it was wonderful. But I mean, I learned a lot, actually. Probably I'm the most benefit person from the lectures. I'm not sure that it's true to participants, students from, particularly from southern countries. The reason is that, I mean, this is what I heard from participants and students from southern countries. There is a big knowledge gap between lecturers and students, particularly from southern countries. They never have any experience on state of art knowledge and research experience, so that they even don't know the terminology. So this lecture tried to narrow down the lecturers' knowledge and the students so that my lecture will help to listen to the other lectures and you can better understand the lectures follow, mainly multiple equilibrium-related lectures. OK, now I'm going to start with systematic approximation of the governing equation. I strongly encourage you to read Holton's book and Petrovsky's book. The governing equation, the set of atmosphere with various complexity from most complete set of equations, this one, to simplest possible equations, I introduce here. Starting from exact equation set from the laws, the first one is Newton's second law of motion field, which one, this is one, the acceleration by forcing. And atmospheric forcing is from pressure gradient and gravitational force. And then maybe you can add friction. And this is thermodynamic equation. First law of thermodynamics, so that heating increase the internal temperature and volume increase. With the ideal gas law, the thermodynamic equation will be like this. And mass conservation continuity equations, convergence change the density, and the ideal gas law. This is a complete set of geophysical fluid dynamics. Now, this contains all the waves, sound waves, gravity waves, loss waves. And if you treat loss waves and rotation, we have to add clear force here. But I eliminate rotation here. And the vorticity equation without the rotation is like this. I'll just show this in order to compare simplest possible non-divine vorticity equation later. This is a vorticity equation. And now, this equation is, I mean, even though it is simple, but it is very complex. So that we usually simplify this complete set of equation to primitive equation sets. The assumption is, this is one single assumption. Lang scale is bigger than depth scale. This is just one single approximation, which gives a hydrostatic approximation. And then pressure coordinate. So now, in the pressure coordinate, pressure gradient is simply expressed by Gdj, geopotential height variations. But here, there is a negative sign here. But P coordinate and Z coordinate direction is opposite. So there is no minus here. So this is the momentum equation, uv equation. I added clear force here, because a length scale is large and time scale is longer so that we have to add a rotation effect so that clear force is here. And vortical momentum equation becomes hydrostatic equation. Very simple hydrostatic equation. And some of the equations are same. And then continuity equation becomes diagonal state equations. Because we use pressure coordinate, omega is d over dp, so that continuity equation becomes diagonal state. So omega is obtained diagonally in this primitive set of equations. Here, the vertical momentum equations is diagonal state. And the continuity equation becomes diagonal state. And the ideal gas law is same. So this set of equation is primitive equations. And most of the general circulation model and the regional model use this set, primitive equation set of equations. So please remember, our all model, general circulation model, mostly probably all IPCC models use these primitive equations. Now, we are going to simplify one more. Density is constant. So that length scale is bigger than depth scale, but density is constant, which is a shallow water system. Shallow water system is shallow, depth is shallow, so the length scale is large. But density assumes to be constant. I think this is a reasonable approximation for ocean water. And if you consider atmosphere, if you consider one layer, average, tropospheric average, I think this is also applicable. So even though shallow water equation is a biotropic single equation with a density constant, and which gives a simple equation, but assumption is reasonable with a density constant. And because density is constant, hydro study equations becomes this one, rho becomes rho constant. And then if you integrate the shallow water equation like this, and p0 here, and height j. And so that pressure is the p0 plus height minus j. This is the mass, and then p0. And if you consider the pressure gradient between these two, and the pressure gradient between these two. It's same, because this much of water is same. So the gradient, the difference between this pressure, and this pressure, and these two pressure gradient is same. So that pressure gradient actually independent of height. So this is what I talk about below. So pressure gradient is independent of height, which is very important. Forcing is independent of height, so that horizontal wind is independent of height. So the equations becomes horizontal equation, u, v equation. And then it can be expressed in terms of height, if p0 is constant. And the continuity equation from here, this one, the exact equation, density is constant. So horizontal divergence should be 0. So even in jet coordinate, this is continuity equation. So that are three equations. Now, because density is constant, ideal gas rho p equal to rho RT, density is constant. So that temperature is simply proportional to pressure. And pressure is proportional to height. So that we don't need. We don't need a temperature equation. If we express, I mean, if we have a height equations. Now, we obtain height equation. And this one, we integrate this very small one, du dx, integrate from 0 to height. Then height is independent, u, v is independent, so that this one is simply h. And if we integrate this to dg, this is w height minus w0. But w0 equal to 0 in the surface, so that this is w. So that wh, top, variation at the top, wh is h. And this height variation, top, is dTdh. So that we obtained the height variations. So as I said, we have a height equation, equation for height, and u, v equation. So that three equations, and three are known. So this is a shallow state equation. So this one, 12, 13, and 14, these three equations. Please remember this shallow state equations. Most of theoretical people use this shallow state equation. Enso theory and MGO, many theories, they use this shallow system with a heating field here. We can add some heating here. Height can be enforced by something. So all these three equations. Now, if you apply call, call means that du, dt, dv, dx, minus k. And then add them together. So operating the co-operator, apply co-operator, then you obtained this equation. So you apply co-operator, these three equations, and you obtained this bodice equation. And you can eliminate this divergence in these two equations. You obtain this equation by combining these two. So we obtain very famous potential bodice equation, dt, dh0. I drove that within maybe 18 minutes, this equation. Now, in many theoretical people, we use this potential bodice equation. Now, what is assumption for that? Only assumption is length scale is longer than depth scale, which is a very good approximation. We are considering the length scale in the atmosphere, much more than 1,000 kilometers. But depth scale is 10 kilometers, so that L is much larger than depth scale. It's a very, very good approximation. Planetary scale waves, I mean, horizontal scale is 10,000 kilometers, so that very good approximation is the hydrostatic approximation. And then density is constant. There's two assumptions. There's no assumptions, right? Just two assumptions. So hydrostatic approximation with density constant. So these equations can be applied to many, many theoretical papers to understand things. Now, here, height, height variations. We have height variations. But if we have a rigid top, just top, height constant. We can make a height constant in the channel with some wood constraint on the top, so that there is no variations. So we call it rigid top. Then h is constant. Then what was the equations? Absolute vorticity is conserved. These very simple equations we obtained, rigid top. In the Charlotte system, if you consider very, very planetary scale waves, the rigid top condition is not bad, actually, because the variation is not big. So this rigid top condition is particularly for stationary wave has many benefits. Even though, from this equation, stationary wave condition actually converted h is constant. I will show that later. So we use this equation. This is an absolute vorticity equation. People think, oh, I understand loose waves based on this equation. But this is now a starting point of my lecture. Actually, in my university, I taught for 30 years wave dynamics. And I teach the knowledge behind this for two months. So everybody doesn't believe. When I start my lecture, it will take two months to understand this equation. People don't understand. But actually, there's lots of material to teach for just this equation. So please, you can use these equations in many, many places for theoretical papers. And people think, oh, this is very simple equations so that this is theory. And it cannot be applicable to observation. So complex phenomena outside. But I will show that many, many things can be explained with these simple equations. Yesterday, John showed many, many good regions based on simplified concept of general circulation with simplified configuration. He showed many good interesting regions. But he still used the same set of equations, same model. But here, today, I'm trying to even simplify the system itself so that with simplified version of equations, I will try to explain several things. And then I will talk about multiple equilibrium. Even this one, this equation has non-linearity. So this equation is difficult to understand. And actually, these equations, people think, oh, these equations cannot be applicable to our weather or climate. But believe me, ECMWF, when they start, they use this equation for extended weather forecast. And the reason that I know was I was weather forecaster when I was Air Forces as a captain from 1976 to 1980. At that time, my responsibility is to produce extended range forecast. But I mean, seasonal forecast. At that time, there is no tool. So there is a statistical model, but nobody believes. I don't believe it. But at that time, ECMWF produced extended weather forecast and ANSEP and JMA so that we received their regions. But extended forecast was based on this equation, biotropic for this equation at that time, not general circulation model. But later, they developed general circulation model, and they start climate seasonal forecast. I think operationally, late 1990s. So this equation is still applicable to many things. Even operational center use this one. And if you simplify this equation, these equations, DTD-botistic plus DTDF equal to 0. And this one, DTD-botistic equal to D-botistic x, dy. We only have horizontal field. And we can linearize this. I will be very kind in the very beginning. So we can make a botistic equal to botistic bar plus a botistic prime. And u equal to u bar plus u prime. v equal to v bar plus v prime. But v bar equal to 0. I'd like to introduce string function here. Because we have a non-divergent condition, because the height is constant so that here equation height is constant in non-divergent condition. Then horizontal divergence equal to 0, this one. So that the divergence equal to 0. Horizontal divergence equal to 0. So that we obtain this equation. Now, this is two unknowns and one single equations. We can make the same equations with two equations. u equal to minus d psi dy. And v equal to d psi dx. Then if you apply this one to here, and this one is to here, it should be cancelled out. So it should be 0. So that these equations guarantee this non-divergent condition. So that we can introduce string function like this. So if you take zonal mean, zonal mean, it must be zonal. d zonal mean dx should be 0, uniform d dx. So that it should be 0. And then applying this to there, and then this one is df dt plus u df dx plus v df dx. Corral force is independent of time, independent of x. So that we need this compound. So in the better plane, this is better. So that beta v. Applying this, I'm not going to derive this. But the equations becomes dt d u bar beta v prime. So this is the equations that I'm going to use in this first lecture. So this comes from the absolute vorticity equations, conservation of absolute vorticity equation, this one. But we linearized. So that this equation is the linearized non-divergent vorticity equation, which was used by Rosby. So this is the original equations that Rosby used in 1938. You better read original paper. Rosby, 1938, in deep research. Anyway, 1938. So he used this equation. Now, let's consider which wave is here. We have assumption that density constant. So that now it gives incompressibility so that there is no compression. So it eliminates a sound wave. Sound wave can be eliminated. And even primitive equation, there is no sound wave. And gravity wave can be eliminated because of the height becomes constant, top. Height or variation gives gravity wave. But we assume that height is constant so that it will eliminate gravity wave so that it only contains rotation change. So the vorticity change by this relative vorticity advection and planetary vorticity advection so that some rotational advection changes the local vorticity. So only the loss wave is here, which related to the rotation. OK, now we have to solve this equation. I will be very kind. And the reaction is pretty kind, in some sense. Now, because of this, the vorticity is minus du dy plus dx, which is gradient stream function. So this one, we can write dx squared d squared plus dy squared d squared stream function plus equal to, oh, no, no. Sorry, plus, I'm sorry, equal to 0. Plus u bar dx dx squared d squared plus dy squared d squared stream function plus plus better d psi dx equal to 0. Now, we try to solve this differential equation. We know that it gives general solution because u is constant. Next lecture, I will introduce why it depends on y, u bar. But here, u is constant. So there is no dependent in the parameter here so that stream function can be expressed like you say very often like this. OK, this is a differential equation solution. I think you should know this. But I mean, everybody show like this. But I think it has some assumptions. We should know the detailed solutions. I think this is like this. k, l, omega, each has amplitude. And sigma k equal to 0 to infinity, l is minus infinity plus 2 infinity. And omega may be minus infinity to infinity. And then, real. This is real solution, x, y, t. Later, I will introduce this solution without knowing this full equation, you never solve other follow equations. So that this means that all the Fourier components is the solution of this. But this is a linear equations so that each independent wave gives independent solutions. So that we just solve one single equation, then we can generalize other solutions as well, other wave components as well. But it really means real components. This is complex, but only real components because stream function is real variable. So this is the complete expression. But as I said, we can express, like psi 0, exponential i kx plus l y minus omega t. It's because this is a linear system. So that if you solve one single wave, then we can generalize all the waves. And if you apply this, and then with some manipulation, we obtained this equation. And this is a loss wave. And equation 2, omega equal to u bar k minus k square plus l square plus k. If you apply this to here, and then with some manipulation, we obtain this. This is this portion relationship. Now, this is a general solution. This general solution can be a general solution for many, many waves, many, many equations, different equations like dx square, d square psi plus dy square, d psi, dx, any equations. Maybe psi, dx, d equal to 0. This is completely different equation. But the general solution is same. But the dispersion relationship is different. So that this actually characterized the wave for this component. OK? Now, so we talk about this dispersion relationship. And then the phase velocity is c equal k omega, which is u bar minus k square plus l square beta. And then group velocity is different. This is phase velocity. If you consider phase velocity constant, phase velocity is 0, which is stationary wave. Stationary wave case, c is constant. But group velocity is shown here, shown here. This is differentiate of the omega with respect to k, is group velocity. I'm not going to talk about group velocity concepts. But even though phase velocity is constant, group velocity can exist. Because for example, if there is a forcing, some forcing, then loose wave can be generated like this. So that if you integrate barotropic potency equation with forcing, with one month's integration, there is some stationary wave like this. So that with some dissipation, the stationary wave component remains as it is for many, many day integrations, the same. So that phase velocity is constant. But in order to maintain this wave, energy should go from forcing reason to there. Because of damping, in order to balance damping, so the energy flow is still there, even though for stationary wave. So the group velocity can exist. Group velocity is the velocity which transport energy. I'm not going to talk about group velocity in detail. But you should understand the difference between group velocity and phase velocity. And with some manipulation, you can drive this group velocity. And for stationary wave, C equal to 0 with this relationship. I will heavily use this relationship in the next lecture, this one. OK. We obtained group velocity as this simple equation, Gx, Ga. Vector is k square plus l square square to vector k vector. Means that Gx direction group velocity is proportional to k. And Gy direction is proportional to l. So the slope, the propagation slope, Gy over Gx is simply l over k. I will talk about this later for the stationary wave. So the energy flow, you can easily estimate the energy flow. And for stationary wave, because of stationary wave, u bar minus k square plus l square beta equal to 0. So that k square plus l square equal to u bar over beta, which is constant for the stationary wave. So if you express this relationship in k and l space, this is simply this circle. Here is u bar over beta. So the wave number for the stationary wave is only this wave, these waves, stationary waves. So in the notation, k is only positive. So this is wave selection. If you know u bar, we automatically notice the wave selections k and l. So other waves cannot be generated. In grocery waves, only kl combination is determined by basic state. And if you have a mountain forcing in x space, x and y, mountain is like this. And if you convert this in Fourier transform in k and l space, because of x direction, x extended is large. So that wave component is small. So only this component, the wave distribution. If you convert this into the Fourier transform, and then each wave number amplitude, if you plot here, this is something like that. So that if you plot this overlap here, for example, we know that this forcing excites the loose waves. Then only loose waves from this forcing, only those waves can be excited. So that even though we solved the equation in general solution and then a linear approximation and then obtained the general solution, but all waves cannot be excited. Only those components, stationary components, can be excited, but with forcing, the possibility existing in the nature is only this one. So that very narrow wave number can be selected. So that if you look at the map, monthly mean chart, global map, the pattern is very simple, wave number 2, 3 component. It's because of stationary wave can be selected only these planetary scale waves. So this is a wave selection, which is determined by forcing, of course, forcing generate waves, forcing, and then basic state, u bar. So that two factors select the waves, and those waves can be generated. And this component, L is negative means propagating south, and L plus means propagating north. And then if you make some manipulation with these two gx and gy with this condition, we obtain this. gx square plus gx minus u bar square plus gy square equal to u bar square. With some manipulation, you can easily obtain this equation. So that if you plot this with gx and gy, this is the geographic pattern of these equations, and this is u bar. So this is the group velocity distributions. So group velocity means that wave is starting from here, and it is this force like this. And this force is to eastward direction, based on u bar, of course. And after one second change, the particle displacement indicates by this gx and gy. When the particle starts from here, it's suddenly disposed like this. But as I said, only these waves can be selected. So that waves probably this much can be selected. So that wave of the particle is displaced this much in one second. And next time like this. So that a wave can propagate like this. But among these waves, only some waves can be selected by forcing. So that this is wave selection for the stationary wave. But even though you can generalize with time-dependent loss waves, the pattern is more complex than this. But we are mostly concerned about stationary wave, so that this is the basic concepts of wave selection and the wave dispersion. So I talk about this. And then the next one should momentum equation. I will spend about five minutes more and then start. Should momentum concept was introduced early 1980s in the literature by UK Lagrangian concept. Wave dynamics like Andrew and McIntyre and Isaac Held. And they introduced pseudo-momentum concepts. I will briefly introduce pseudo-momentum concept. This is very much introductory concepts of loss wave dispersion. Now, pseudo-momentum concept, which is wave activities. So we have to know how the wave activity looks like and then how wave and wave flow can interact, can be obtained from pseudo-momentum concept. I'm not sure whether we still use these pseudo-momentum concepts. But I learned this kind of concept from Isaac Held when I was a GFDR in the late 1980s. Anyway, the equations that I use was the simple equations. This perturbation equation, very simple equations. We just multiply by this. Then we obtained dt, the zeta square. And u bar dx, the zeta square. Plus beta v prime zeta prime equal to 0. Let's take a journal mean. Take a journal mean. dx, the journal mean will be 0. So that dt, and then we divide by beta, beta square equal to minus v prime zeta prime bar. So that this is wave activity. Actually, later on I will show that this is the momentum scale. So that we can talk about this pseudo-momentum. But in many literature, they also use the wave activity. And so the wave activity changes depending on the journal mean with the activity change due to bokeh's direction. Now this term is simply as dy v prime u prime bar. This v prime zeta prime bar equal to dy v prime bar is I draw there in this box. Please read this one. Here, I just add here. This one, I added this one, this component. This is because u prime dx d u prime equal to dx d 2 over d prime square bar. So dx, the journal mean should be 0 so that you can add this. Because this term is 0, so that all this becomes 0. So that this direction is equal to add momentum components. And then dt d u equal to minus d phi dx minus plus f v. Now if we take journal mean, v journal mean v bar is 0. So there is 0. And dx d bar is also 0. So that d u bar dt equal to 0. This is what I talk about here. This one, all this equal to 0. And then if you manipulate equations, we simply obtained equations dt d u dt d u bar equal to minus so u i d u prime v prime bar. This is I draw here. So we can combine these two. We can combine these two. So that this is same. Same as this one. So all together is dt d u bar plus e to the power equal to 0. This is conservation. So this is wave activity. I said should momentum or wave activity. Wave activity plus u bar equal to constant. So this is total momentum, any momentum activity. And journal momentum, total momentum should be conserved. So that when u bar increase, so that at the activity, journal momentum should be decreased. So this is the wave mean for interactions and kind of angular momentum conservation. Total angular momentum conservation. Can you do the last step to go from the bar and d u bar dx equal to u prime v prime bar. So same. OK? Same. Maybe I think this is plus so that it can go. So same. This one. And you can substitute here and you obtain this one. So with a few minutes break, maybe five minutes, maybe with some break, five minutes break, I will talk about this. And I will talk about barotropic instability based on this equation. OK, thank you.