 Twirling latitude is so important when the wave propagates. When it's turn, it's wave amplitude is maximum, very big amplitude. P&A를 제안하는 것입니다. 밖에서 기술을 연결하고 있는 기술이 있으며 기술의 아낌을 제안하는 것입니다. p는 높은 랍지척입니다. 왜냐하면 l는 높은 랍지척에 적용됩니다. 높은 랍지척은 높은 랍지척입니다. 여기요. 이제는 크리티컬 랍지척입니다. 크리티컬 랍지척은 여기, 스테이지를 통해 k스퀘트와 l스퀘트는 u과 v과 v이 더 좋은 것입니다. 이게 스테이지를 통해 그런데 station wave, station wave, 이것은 이러고, y-square, z-square, z-square, z-square 여기 스테이셔널리 웨이브 넘버의 L스퀘어는 minus K스퀘어입니다. 그리고 스테이셔널리 웨이브 넘버의 C of beta-9입니다. 이건 스테이셔널리 웨이브의 케이스입니다. 그리고 이 웨이브는 스테이셔널리 웨이브입니다. mount of the critical attitude is when u bar is equal to C. If this number is equal to zero, then L is infinity. Actually, the solution is breaking down when u bar is equal to zero. 전체적인 경우, u과 equal to 0입니다. 이는 크리티칼라티드입니다. 그리고 이 숫자는 infinity입니다. 그래서 l는 infinity입니다. l는 infinity입니다. 그래서 크리티칼라티드가 뭐죠? 크리티칼라티드가 어디에 있습니까? Since l는 0을 equal to 0, which is here, the soft tropics, like about 10 degrees here, u과 equal to 0. This is the critical attitude. About 10 to 15 degrees, u becomes 0. So that wave number l becomes infinity value. So that you cannot define l when u과 equal to 0. Now, because l is a very large number, and group velocity, which is defined here, group velocity, l square, l is cancelled so that it's proportional to l triple, 1 divided by l thought. So that l is infinity, so that group velocity, y group velocity is 0, almost 0. So that when a wave approaches to a critical attitude, I mean the group velocity, y group velocity is almost 0. So that it never reaches to a critical attitude. So turning attitude is the attitude when the wave is turning. Critical attitude when a wave propagates, and it never approaches to a critical attitude. means that a wave cannot propagate close to a critical attitude. So it is interesting that we have a critical attitude here and here. means that wave is generated in the extra tropics, and never pass to other hemisphere. So it is always confined in the extra tropics between subtropics and higher attitude. And here is turning attitude. Here is critical attitude, turning attitude here and critical attitude. So that wave is confined over here. This is very important so that one hemisphere, wave generated, the wave generated in one hemisphere never influence to other hemisphere. So this is also important concept related to free waves and critical attitude. And force loss wave. Now we are using height cannot be constant. Because mountain or topography height is varying. Let's talk about, and previously I made a constant. So that absolute vorticity is conserved. But potential vorticity height is also varying. So that there are so many ways vorticity rotation can be generated by change of the critical parameter, also change of height. Height change, then it creates vorticity. And height is at, and this is height. And there is topography, topography. Then the height is from here to here. So as I said, h is height. This is height. Height is h0, h0 is the mean height, h0. Plus eta minus eta b. This is free surface. If you do not like free surface, it can be constant. Then previously we talked about non-divergent concept. But for a while I like to keep free surface. So the one equal to h0, 1 plus h0, 1 eta minus eta b equal to 1, which is h0, 1 minus h0, height minus height b. So 1 over h is this one. So multiply by vorticity, zeta plus f equal to 0. Same thing, same equations, this one and this one. So using this, height variations will create vorticity. This is the equations here, this one. f0 plus beta y is this f0 plus beta y, this one. And because f is the biggest quantity, and if you linearize these equations, we can simply queue these equations, as I said here. And you can easily drive this equation. And then the linearized equations, because this is vorticity's concept, and this is the q, dtdq equal to 0. And if you linearize this, dtdq plus u bar dxdq. star means that wave component, maybe this one. If you linearize this, plus beta, maybe this notation, beta, dy. v star dy dq bar dxdq. This is a linearized equation. You can follow these equations. So based on this equation, we can drive topography force waves. So this is about. And then q bar is mean state. And this is perturbation. And only f0 is the journal mean correlation force and beta y. But perturbation with respect to this one, so that vorticity square and eta star minus topography. And you easily obtain this equation, equation three. I'm not going to talk about the detail derivations. And now, and finally, we obtain equation five. This equation has been used by many people, based on prescribing the topography. And you can generate it at a vorticity field. Stationary wave case, this term is gone. So that it goes to the original non-divergent vorticity equation forced by topography, because this is gone. And then you can obtain by prescribing topography. You obtain the stream function. And this is the stream function that Isaac Held obtained in 1983. And this is topography. This is topography, the Himalayan mountain and Rocky mountain. Actually, this solution is based on global model. Globally represented this one. And then this is the model results. And this is observation. So that even this is very simplified equations. You can reproduce planetary scale wave very nicely. So that this equation is actually not bad to generate the stationary wave along the external tropics. As I said, this equation was first used by Chani and Ariasen. And since then, we tried to use that model for prediction purpose. And as I said, until recently, we used barotropopatistic equation for seasonal forecast, monthly forecast, until late 1980, even early 1990s. So this equation is not bad to reproduce observed stationary waves. Now, we can introduce the wave solution like this. Because here, u bar is constant, beta is constant, so that you can use this waveform. Topography also has form like this. And then you substitute these two. And then you obtain this one. And omega can be defined like this. You can easily obtain it. I hope you derive all this equation. And this is your homework. And I ask you to read this textbook, my lecture note, five times. Then you may be different. You may understand many things. Please drive everything, all the detail. Because I draw in this lecture note. So please follow everything. Now, omega equal to 0 is stationary wave. So that ks is simply k-square plus l-square is beta of u. So that stationary wave selection is same as free wave. k-square plus l-square equal to beta of u bar. So that stationary wave selection doesn't influence by mountain itself. So in the above, a system cannot provide stationary wave for negative journal mean flow. If negative journal mean flow, then this one and this one is all negative so that omega cannot be 0. So only stationary wave can be obtained with this wave numbers. So stationary wave number selection is determined by the basic state, nothing to do with a mountain. And topography of force wave is here. You can obtain here at hat obtained like this. You can follow this equation anyway. So here, eta is wave amplitude, which is proportional to height. height will generate wave amplitude. Not only that, but also this k minus ks squared. So which is determined by this ks squared? k is determined by the topography. Because topography defines the wave number. So that k is determined by topography. And ks is determined by the basic state. So when topography wave number is same as stationary wave number, it is mountain resonance waves. So that wave amplitude is very big number. So that mountain resonance wave, mountain resonance wave, which we can see outside. But if you have some damping term, which is this one. If you add some damping, damping term, it becomes singularity can be eliminated. So topography force wave is very much depending on the basic state itself. Basic state, basic state. If you have basic state, do not mean flow. Oh, we can diagnose many things. Not only topography force, free waves, many things. And the wave behavior like turning latitude, where is the turning latitude, where is critical latitude. And many things we can diagnose. And then eta is the wave amplitude. So that the final solution is like this. As I said, real. And this is wave amplitude. And this is phase. If you define height, topography as a function of a regional height as a function of sine curve, it becomes like this. So I'm going to use this wave solution. Topographic force, loose wave solution, this one for drive multiple equilibrium. Understand, it's a very important concept. Wave resonance topography, which actually creates the stationery wave that we can see outside. And multiple equilibrium now. We can drive this equation from here. I actually drive this equation from potential potassium equation. As I said, Q is F0 plus beta y plus potassium. And this topography term. And with journal mean, this is simply this. If you make a journal mean, you can easily obtain this one. And then following these equations, you can obtain dt dy by cancelling dyd here, cancel this y. So you can drive from here, you can get this equation. OK. Now we have a journal mean equation, which is affected by a wave. 2 terms, u bar plus equal to v and plus. And then we can add some damping term. Damping term minus k u bar minus u equilibrium. OK. So previously, I talked about this, free waves. This is same as dyd v prime, u prime. OK, journal mean. The same as this one. I already talked about this. And then this is topography first forcing. Now the journal mean flow is affected by topography. How it affect topography is like this. Topography is like this. And then wave case is the pressure. If wave determines the height, maybe high and low, if you have like this, then pressure is like this and pressure is like this. So that in the topography, in the same altitude, here in this case, there's some pressure force. With respect to here, pressure here is more, so that it spins. So that all is rotating like this. So that lativeness, relative angular momentum is this way. So that this is the way that journal mean change by topography. Otherwise, if there is no topography, it is all canceled. Plus, minus, plus, minus, all canceled. So that there is no angular momentum change. But if we have some topography, there is this continuity here. Which makes angular momentum change with respect to all rotation. So this is this term. I can drive this concept from this. But because of time constraint, I'm not going to talk about this. But anyway, this is the form drag. And this term is important to change this one. And I think I already mentioned that this doesn't contribute to change this u bar. This is here. Here. v star, v t star, v is dx zeta. And v t is k square, second derivative of eta. So that dx zeta square multiplied by zeta square is dx zeta square. So dx zeta square, journal mean of dxd equal to 0. So this one, actually, this term, this term, vorticity flux, or momentum convergence, actually doesn't contribute to change of the journal mean. So that this is kind of non-acceleration concepts. So free wave, free addy, never change mean state. If free wave change mean state, then basic state is changed with respect to a wave so that how we can fix journal mean. But this term never affect mean state. So that it doesn't contribute, eventually it doesn't contribute. So that these two can balance. And these two balance has two state. This is what I'm talking about. So you have to derive this term, this equation. And as I said, eta equal real number. This is already a draw here. eta is this one. So real component of this, just multiply by k square minus a square plus i epsilon. So that this one and multiply by this one. And real of this component, as I said, real of a equal to 1 over 2 a plus a conjugate. So that you have to make this so that you obtain these equations. I already wrote this equation. Real multiplied v square multiplied by ht mountain, which is an expression, one over half of this one of conjugate of this. And this one conjugate of this. And then if you multiply, this multiply by this two multiplication, we obtain this one. So that form drag can be expressed like this. Now let's discuss this. This form drag, this mountain forcing to journal mean, actually determined by journal mean state. Very much journal mean state here, journal mean state. And ks is also determined. k is determined by height. It is excited by topography. So that when topography is given, then k is given. So that k is determined by topography. And here is topography. But basically, u bar. And this term is very important. Topography, the resonance with a basic state is important term. So that this can mean topography is given. So that we can draw this with respect to journal mean state. So that this is the line, this is the curve for this with respect to journal mean. Topography is given. So that we can plot this. And then we plot this one. So that this one and this one is plotted with respect to u. And then cross-section is the solution. So that this is a way of obtaining solution graphically. So that these three intersect point are the solutions. Now, these two are stable solutions. But this one is unstable. I will talk about this. The stability. Although the argument is based on a linear wave response to given journal flow, China and the world showed a multiple equilibrium state in the numerical solution of the model in beta plane channel. So that these two, the stable solution is obtained by numerical solutions of this barotropic potassium equation. Now, state A. Here, state A. This term is, we plot this term. And this term and u, this is this term. And the difference between this is journal mean change. Now, when this is larger, when journal mean becomes larger, this term becomes larger. But it is smaller. So that the difference actually negative. So that when journal mean is increasing, then the tendency gives back. And if it is larger, then the tendency becomes positive. Because you can easily see the difference. So that it goes back to the original state. So that it is stable condition. But in this case, it is unstable. So that we cannot choose this one. And this one also can be a stable solution. And you see, there is two journal mean state can exist. This one is high index case. This one is low index case. So this multiple equilibrium is based on the interaction with topography and wave journal mean damping. And this term and this term is competing. And which create two solutions. And China and the world also shows in the numerical solution that they two solutions can exist. But there are many, many arguments about this multiple equilibrium. Multiple equilibrium theory that I show here is very beautiful. Very straightforward, nothing difficult. And you can easily derive this equation. Because I stated all the equations there. Without this kind of concept, you cannot derive these equations. So if you follow this lecture note, you can easily understand multiple equilibrium. But the existence of multiple equilibrium. This is based on linear theory. So that if we add nonlinearity, the solution may not choose multiple equilibrium. And I think you may heard about KK Tong. He was in MIT working with Chani and Rinsen. And he is now in University of Washington. He is a private mathematician. And I had many chance to talk with him. And he said that with very careful examination of Chani and Dibble calculation, there is a mistake. This is what he told me. And this multiple equilibrium is questionable. But theories are that multiple equilibrium theory is beautiful. And in linear theory, multiple equilibrium can be supported. And Franco-Molteni, Franco-Molteni 사실은 multiple equilibrium state using observational data. What he did is, I think in some years, I don't remember the paper. Actually, he made any statistics, maybe temperature field or velocity field, whatever. And this probability, and this is the temperature field. Then usually, we have a normal distribution like this. If you make this probability distributions, you can sometimes obtain like this. OK? So that means that the state doesn't have the normal distribution, but more chance in this region. So that by saying this, they argue that multiple equilibrium can exist. Nature paper or quality final answer? No, Molteni forced us a paper. I don't remember. Maybe a nature paper. Anyway, he demonstrated using observational data. And yesterday, John Marshall also talked about multiple equilibrium state using their model. So complex model, which gives multiple equilibrium state, which may be different from this one. But basically, the concept is same. So multiple equilibrium is really not settled down at this point. So that I don't know how you choose multiple equilibrium as our title of this summer school, Ikado. Why did you choose multiple equilibrium? I didn't discuss. It's a long story. Long story? Actually, long story. You may talk about. But anyway, our summer school is about multiple equilibrium. And I talk about multiple equilibrium theory. And I hope my lecture actually help you understand other lectures following. And I introduce all the equations and theory about waves and mean flow interactions. So my lecture is nothing to do with state of art research results. But I try to teach the basic knowledge about. And I prepared about 10 minutes to talk about something. Please. Now I'm talking about some mental lecture for 10 minutes. Since you are students and you are the person who try to be good scientists, in order to be good scientists, your environment should be good. As I said here in the wave theory, basic state determines everything. If you know basic state, we already know what the structure is of the wave. So particularly from a southern country, you should improve your basic state so that you have to interact with good scientists more. In order to interact with good scientists, you have to demonstrate your ability. So that try to get the opportunity to interact with good scientists by demonstrating your ability. I will talk about two things. Maybe three. Yesterday, I wake up very early in the morning. I thought about what I'm going to talk about you. And I certainly remember two things. One is my experience at age 15. At that time, Korea was a poorest country in the 1960s. I mean, at that time Korea per capita is about one fifth of the Philippines. I think now about 10 times more than the Philippines now. Anyway, very poor at that time. But education at that time, education was very strong, very, very strong. And all the parents tried to push children to work hard to study. And at that time, one big thing in high school was math competition, mathematics competition. And the number one student becomes a national hero after math competition. And I know at 15 years, people usually go to math competition at age 17. And third grade of high school. And I know one person. And after the competition, there is an announcement. Oh, I was surprised. The guy, I know. That was the first math winner. And everybody surprised. I mean, at that time, usually we predict who will be number one. But he was a completely unexpected person. But I know him because I lived close by. We lived close by. And I know him very well. So a few days later, I talked to him. How you become number one? And the national hero. He even doesn't understand. He never predicted. He is very poor person. His father passed away a few years ago. And he lived with his mother. And very poor. So that in the math competition, he doesn't have a pencil. He doesn't have a laser. So that he asked his friend to borrow a pencil. So I mean, he was number one winner. So he said he doesn't have any money. But he somewhat, I mean, one year old at the time, he got some mathematics book. Mathematics book. He never thought he thought. But he has one book. And he has no choice to read other books. Just repeat read one book. And probably more than 10 times. Just one book. But that book is not popular at the time. Nobody knows. But he has only that book. But he repeatedly read this book. And then he become number one. And then he gave me this book. Very, very. I can even recognize the page. Because he read many, many times. So I think this is the way that you have to concentrate. You should repeat, repeat, repeat. I mean, probably you took 100 courses. What do you remember? When I select the participants in my summer school, I don't select the person who has many training courses. Don't be trained. Don't be trained. Demonstrate. And do, I mean, repeatedly work one thing. This is one thing that I talked to you. So that I gave you very kind reaction. Please read this more than five times. This is one thing. And second thing is you should demonstrate yourself. This is one of the first cases about my two-year senior person. And now, second story is about me. I actually involved in ICTP 30, when Shukra start, I was here. So that I have a long history. But 15 years ago, I don't know, I stayed here for one month. Probably Shukra asked Franco. Franco Molteni was head of the section and Fred was here. And at the time, I happened to stay here for a month. And I really wanted to come again and again for such extended period. And in the first week, we had a meeting. Fred, Franco, and some of the people talking about answer predictability, answer predictions. And at the time, they developed a speedy couple with MITGCM. And Aliza did that. And we talked. But I suggest Franco. Oh, Franco is a speedy model. So that we better couple. That we intermediate couple model. Okay? Like Ken and Ji-byung model. So, Inseq, can you do that? I said, okay, I will do it. And then I communicate with my students every, I mean, all the night. And then within one week, I coupled speedy with intermediate couple. You remember, right? Franco was amazed. And then I brought some good results. And I gave some surprise to him. And then in the last week, I talked with Franco. Franco, I'd like to come again. He said he immediately offered a stable associate title. So that you remember, right? Stable associate. Which means that I have a right to come again, again. And then he brought the draft. And Inseq, it's okay. I see this. I have a right to come one month every year. So I said, oh, one month. I like to enjoy two months. And I didn't know much. So I asked Franco, please change to two months. He immediately changed. I mean, Franco is very slow guy. He never fast like that. He immediately brought that. And I signed. He signed. So this is a story 15 years ago. And repeatedly, I mean, I always expect to come to here. And I really enjoy. Maybe saw the story about Ricardo. So that demonstration and desire is important. And you should demonstrate yourself. And we had the teleconnection workshop. When was it? It's the year 2008 or 2009. Very famous teleconnection workshop. Nine, maybe nine, maybe nine. Very famous teleconnection workshop. And Fred and I organized teleconnection workshop. And at that time, Ricardo was working in GFDL. And Sarah and Ricardo came together to teleconnection workshop. And in the first day, he said he arrived in the night. And then in the next day, he just opened the curtain. Oh, it's wonderful. So he was amazed. So then he decided. He decided. He's Italian. So then he decided to be hired by ICTP. His desire. And he made effort. And he become a staff. But that's what you told me. But this is an initiation. There's many reasons. Okay, okay. Anyway, my message is that you have to desire. You have to desire first. You have to demonstrate. You have to demonstrate your ability. Working hard to achieve your desire. That's it. You have any questions? And no questions? Thank you very much. I'll finish.