Non-Linear Dynamics as Framework for Understanding Subquantum Structure — Part I

0:00 ...but this is what I want you to think about. I want you to think of a system which is just defined as flow on its invariant set. I don't want you to think about the fact that this is embedded in some of the Euclidean's space. Now, I'm going to, in the next talk, I mean, there is a methodology for doing this which is called symbolic dynamics, which is part of non-linear dynamical systems that are not a very common thing. But you realise, you know, this is a kind of difficult thing to describe mathematically, if you like, the intrinsic geometry of this type of attractor kind of independent of its embedding. That's the thing I want you to talk about. So if you perturb the system off the attractor, then I've kind of gone off I've gone out of my allowable... The system is no longer part of my system now. I'm out of the system. That's a key point. Because... I might come back to Bell now, again. So, just to recap, I started with a quotation from Bell. And he's making the point that the whole conclusion that quantum theory cannot be described by local causal theory and assumes some notion that we have these three variables, the experimental parameters of three variables. Now, of course, people often say, oh, well, you know, since experimental systems are governed by experimenters, maybe all that's telling us is that experimenters really don't have free will that we're somehow we can explain the Bell paradox by just saying humans, experimenters don't really have free will, we think they have free will but we don't. But actually Bell makes the point that he doesn't really think that this issue, the existence of otherwise a free will for human beings relevant, or you know, this will not explain the paradox of the Bell theorem in his view. And the reason he thinks that is the following. So we come back to some quotes. And it's interesting, I mean, this was in a paper written in the days, really, before

2:30 chaos theory, but it's sort of you can set it in a kind of chaos, chaotic context, because he says look, just to make the point that free will is not the key issue Let's suppose that the instruments, this is our polarizing filters or the Schoen-Gerlach apparatuses, are not set at the whim of us experimenters, but we just set up some mechanical random number generator. So this could be like a chaotic piece of software which somehow runs. and the output of that what he calls mechanical device or care of the system could be regarded as sufficiently free for the, well, he poses the question, could the output of such a random number generator be regarded as a sort of free variable for the purpose of hand? He says, I think so. I think this is reasonable to imagine that. And the reason he says that is this paragraph here, and this is an important one. So, he says, let's imagine that, and this really is like a kind of chaotic system, let's suppose that how you set your experimental orientation depends on whether the digit, the decimal digit, in the millionth place, or billionth if you like, of some input variable, is odd or even so you stick in this sort of random number you don't really know what the millionth place is the system trunters away and its output is actually determined by whether that millionth digit is odd or even so Bell says knowing the output fixes something about the input in other words knowing the output tells you whether that millionth digit was odd or even now this is his quote thinks that this effectively behaves like that millionth digit is effectively a free variable. He says that this peculiar piece of information, the value of this millionth digit, is unlikely to be the vital piece of information for any distinctively different purpose, i.e. it is otherwise rather useless. Can I jump in, Tim? Yes.

5:00 My problem with that is Well, this is Bell. I'm quitting Bell. I know you're going to say exactly what I'm thinking, but my problem with that is that you've still designed an apparatus which has only two settings and that's the possibility of just those two settings is not random and given that you're You're talking about non-locality, ultimately. It isn't excluded in principle that the fact that for a year you've been building this apparatus so that it can have just those two possibilities in a non-local way communicates its information to the system you're going to study. Okay, yes. I mean, you know, the two choices is slightly irrelevant. Well, that's a sort of perspective. I mean, that's, yeah. OK, I mean, my take of... I'll just say very quickly, the point is, which is precisely the thing which Bell is stressing here, is what the philosophers call failure of counterfactual definiteness. It's nothing to do with free will as such. I mean, he's trying to get away from this issue about pre-work. He's just saying, here's a mechanical device. And he's saying, look, that millionth digit, I mean, he's not serving any other, according to his intuition, he's not serving any other useful purpose in the universe other than for setting the output of the orientation of the same guy. In other words, he's saying the universe would function just as happily, in every other respect, if that millionth digit had been even when, in fact, it was off. So, in other words, Bell considers that the parity of that mean digit to be 4, for practical purposes, 3. Now, so far this is kind of, what I'm trying to do is, this is kind of justifying the existence of these three variables in a way which has got, you know, tries to get the human,

7:30 and these kind of philosophical nightmares about free will out of the argument, okay? We're just talking about now mechanics, in a sense. Now, my point is, however, this intuition is actually inconsistent with this Tarkin's theorem that we've never talked about. If one took the view that the universe as a whole is governed by this type of deterministic non-periodic flow on some tracting invariant set. Because you're not at liberty by the Tarkin's theorem to view any variable, even if it's energetically unimportant, as indeed a million places of some input variable would be, you're not at liberty to treat it as an otherwise rather useless variable. Because by the Tarkin's theorem, all you have to do in a variable enough times, and the theorem says you can reconstruct the entire attractor of your hypothetical universe. So, or to put it another way, if you believed that the system is governed by deterministic non-periodic flow, some low-dimensional attractor, then a perturbation like a counterfactual one type of Michael was talking about what would have happened had I changed that digit from an even to another one, it was let's say for the sake of reality odd, what would have happened if it was even that type of perturbation will take you off the attractor and if you take it off the attractor it's no longer part of the system that I have defined to be my system physics I just want to back, so this is the quote again from Bell. I don't want to talk about metaphysics, I just want to say, I want to just analyse Bell's theorem from a certain kind of physical theory. We've yet to see whether this is a viable kind of physical theory in terms of quantum mechanics, but I'm being not worried about that. I want to just consider Bell's theorem analysed with respect to a kind of physical theory in which the state of the universe might be so, and after all, you know, this type of chaotic system and chaotic dynamics, it's pretty generic, you know, there are loads and loads and loads

10:00 of examples in almost every type of science, certainly any type of non-linear science, but chaos seems to play a role. So it's a pretty generic type of phenomenon. With the additional restriction that I want to consider flow on its invariant set or on its tractor, then I would say that this type of theory is objective, it's deterministic, and if you like to use the word realistic, in that sense it's realistic, but it does not admit the freedom to allow these kind of dynamic beyond constraints, what would have been if, types of questions to be posed, i.e. these counterfactual perceptions of what appears to be otherwise unimportant parameters. I.e. experimental parameters cannot be treated as free variables in such kinds of physical theory. Let me just move on. What I want to do, and I'm not going to do this now, so I'm just going to come to this tomorrow, because, I mean, you might say this is all very well, but what's this got to do with quantum mechanics? And can one sort of fit quantum theory in such a sort of framework? And I think one is faced with three problems here. One is how you actually represent mathematically this type of flow that I'm talking about, flow on its attractor. And I don't want to talk, you know, just to talk about the differential equations is not actually enough. I need to talk about differential equations and boundary conditions together. And that actually raises issues about uncomputable representations. And there is a methodology which is called symbolic dynamics, which is a way to do this, and I'll talk about this tomorrow. So I'm going to do a little bit more about this notion of symbolic dynamic representations of deterministic non-periodic flows. The states are basically represented by multiple symbol strings. And if you have, say, two systems, two of these types of systems, where the symbol strings are correlated in some sense, that denotes some type of synchronisation. So chaotic synchronisation is a very important process in chaotic dynamic. which behave completely irregular and nevertheless come into synchronisation if they're so coupled. So that's the first part. The second part is to say how can you reformulate quantum theory so that it might fit into this framework? So the second part of the talk tomorrow

12:30 will be to try to discuss the standard for Hilbert space representation where you can try to think of quantum states as these symbol strings. And complex structure will be, I'll talk about permutation operators which have quaternionic sort of similar structures. Then I want to introduce the other entanglement in terms of these symbol string correlations. And then finally, you know, the perhaps $64,000 question is, the linearity of the Schrodinger equation, how does that encompass in such a non-linear theory? Well, I want to talk about the existence of linear structures in symbol space which can perhaps model this type of so I'm trying to I'm trying to in a sense bring in some new physics to do with non-linear dynamics into Islamic quantitative and my belief is that will that will but is Basil oh it's gone oh you've got the quote because I've had this paper on my desk for a couple of years and I've been meaning to get round to reading it and I've only got round to reading it on the plane I'm just reading it now on the way over to see what we said I've only got round to I'm sort of slightly embarrassed about this but I only got round to reading it on the plane on the way over yesterday you never know with things when you have a kind of fixed you read something and you think oh yeah that's exactly what that really resonates so so this is a paper uh written by david and basil back in 81 uh on you understood in terms of non-linear field equates so they're talking about non-linearity at the time i guess uh the part of renee tom and the catastrophe theory was very much in the boat But in a sense, catastrophe theory is about just bifurcations of nonlinear dynamical systems. So it's kind of part of the whole framework of which kind of chaos is, you know, the general kind of concept. But they were addressing this issue of could we kind of understand exactly the same thing, not an entanglement by nonlinear types of thinking.

15:00 And this was just a quote I just put on the paper. We are supposing that through non-linearity, the movements, so they're considering the EPR experiment where they have this source, positronium source emitting polarised. So this is the source, and then they have the detectors, A and B. So we're supposing that through, I don't know if this is Basil may correct me, but I thought this was a key sort of sentence in the paper, I must say I didn't perhaps read every single sentence but anyway, we're supposing that through non-linearity the movements of the positronium and of the particles constituting the detectors A and B are coordinated in a certain way sort of through this non-linear so the nature of the coordination is such that the positronium atom decays into two protons, intuitively related but well-defined polarisations only when the detectors are in a condition to absorb these photons. This coordination is not the result of pre-established harmony nor is it by a conspiracy. That kind of resonates to me about this fact that there's this overall, the whole system, which is the positronium detectors that make the universe in some sense, governed by this underlying non-linear dynamic, this schematically, this type of this type of overall structure and that that uh in a sense um means that even these dynamically kind of what seem to be energetically unimportant degrees of freedom there isn't there isn't you don't have the dynamical freedom to perturb those in arbitrary ways you take the system in my language you take it off the attractor you take it away from its invariant set we'll see as I go through the talk whether this has the Basel thesis at all just re-reading what you put on earlier that's why I went to get my computer just to check what we actually it was 81 you know what did you mean by conspiracy well that was the whole the whole terror of the debate was whether there was some conspiracy in nature, to fool us into believing it was non-locality when all the time it was locality. I think Vigier had this idea that if you had it all on a tape so the universe was a sort

17:30 of a film which was unfolding, then the correlation would have been put in there, locally, and as it unfolds then the conspiracy is that God's designed it in this way to fool us. That was the, I think, the idea of conspiracy. Conspiracy means literally breathing together, so it's more like what you're talking about. That was the debate we had. I've been gone into it, really. I'm sorry, I just, you know, how is... I think the key, my perspective on this is the greatest word that Michael raised about counterfactual perturbations, that things that, you know, in most physical theories, In most physical theories, you have an initial condition that you can kind of perturb that initial condition and you run your differential equations and everything kind of happily jumps on as much as they did before. But these type of systems, you can't do that. If you say the system is defined by differential equations on its invariant set, then you're talking about this low-dimensional measure zero set in space space, and any perturbation will take you even if it's infinitesimal in the literal sense it will take you off that system and so that becomes now no longer dynamically allowable and in a sense with respect to the measure of your state space any such perturbation you know with probability one will do such a thing and this is as I say consistent with this Tarkin's theorem that any degree of freedom you know even if it's totally energetically irrelevant can be used to reconstruct the entire attractor. It's a kind of mega it's kind of Marx's principle to the extreme if you like. So this is the basis of what I want to develop as kind of a more of a mathematical theory to see, you know and perhaps the apart from sort of trying to let's say contribute to the discussion about the meaning of bell and bell and non-locality what I think I would like out of it is perhaps a new kind of conception of what gravity is. I think I'll leave that, really, because the next slide, I think, goes on to some, you know, sort of mad stuff. So it's half past five now. I could just give you a quick lantern on global warming before I answer this.

20:00 Yeah, that would be really good. Can I just say something very quickly on the purely historical side? I really found this extremely interesting, really, thank you very much indeed. just on a historical side and also a question to Basil I assume that what Basil and David had in mind sorry Basil I assume that what you had in mind by saying no it's not a conspiracy and what you had in mind was that it's the quantum potential this was definitely coming out of the non- this was rejecting This was going right away from this... You can remember, Damon and I used to have different standpoints in which our theories were actually inconsistent with each other if we tried to view them in the whole. Right, that's very interesting. This was to throw the polar potential and all that to one side. To put that aside, to bracket that and look at what it would be if it was just a... I think that in the introduction that they made that point explicitly that they said this has really got nothing to do... With your ideas about other possible sources of understanding of non-locality the quantum potential okay well that's very interesting that's historically very illuminating the other thing i wanted to mention very quickly was that around that time 1981 when they published this paper i don't know if this was at all in your background you know in your minds when you published it uh but since you specifically cited you know einstein i said i did of course in the last years of his life do quite a lot of work on old-fashioned you know non-linear uh unified field theory, so the kind that me and Hilbert had worked on way back before 1915. Sachs had published a number of papers on a non-linear spinner field theory, in which quantum theory emerged as the weak coupling limit of that theory. And that's quite unlike, you know, your idea of basing it on the attractors, but that, you know, there were some, they were very much non-mainstream, but they were in the air around that time, in the early 80s. But that didn't influence you at all. A lot of linear theories which would just produce a steady state or a limit cycle or anything like that is kind of uninteresting and not relevant to this. It's the fact that you can generate completely... You can sort of describe all the irregularity of the real world with these low-dimensional measure zero attractors. And these are the ones where you have these kind of fractal structures to the attractor.

22:30 And that's the thing which stops you from just arbitrarily perturbing states that's what breaks the counterfactual definition that's what breaks the counterfactual definition no I mean your Sarkes was certainly not influenced by ideas ideas about the Lorenzo Tractor or ideas about the hydrodynamics and meteorology at all and I think this is if you're going to try and get down this road your approach I think is a much more realistic and promising way of doing it but I simply cite it because it was an instance of what's always been in the background as a bit out of the mainstream but has been an interesting kind of continuing tradition in physics Tilbert and me, which is that there have been people who have continued to work on non-linear spin and field theories to try to get out... There may be something, as you'll see tomorrow, even from the first slide, I start to use some of the intrinsic counter-set-like properties of these in varying terms of manifolds. Can I just... It was actually, on this historical... It was actually Einstein himself that inspired... Yeah, right. non-linearity, because he was dealing with non-linear equations. And therefore, what David and I were doing in our discussions was could we take Einstein's ideas a little bit further and perhaps Well, Sachs, of course, always claimed that he took it directly from Einstein. He was just trying to consider what Einstein was trying to do. We were independent of Sachs. Yes, go on, go on. Okay. So, just to now for something completely different this is a talk that I've given in a number of places but they've always been to this has always been to a kind of lay audience not to physicists so I'll skip over stuff or you may be just amused by the level of science in this but maybe a few things you didn't know again actually, you know, this was a one-hour lecture, but so what I may do, if I kind of, I'll just perhaps, I may not give the whole lecture now, I may sort of, again, split my talk up tomorrow, because I don't want you to, I know it's late in the day, and if it's me, I'll be fed up after about half an hour. So let me just sort of see how I get on, and if I don't quite get to the end, I'll just keep the last bit, until the last bit will be actually what the predictions are.

25:00 I should say this was motivated by I said this at the Royal Society which is so there was a BBC programme actually which was based on this lecture which involved me hitting golf course on the golf course and this arose because I said and this was a lay audience which included the BBC producer who made this programme I said that this kind of motivation of this talk arose pretty much from a number of golfing partners I've had over the years who said something like you know, you guys can't even forecast tomorrow's weather night, so why on earth should I believe any of this global warming stuff? So, you know, he raised this question well, how can we be sure about these predictions of global warming stuff? Now, let me yeah, so so we start with this kind of iconic picture of the Earth from the moon, from one of the Apollo space shots in the 60s, and with an interesting statistic here, that the surface of the Earth is roughly 14 Celsius on average, and the surface of the Moon is roughly minus 18, so there's about a 32 degree difference. And when you consider, you know, basically the Earth and the Moon are nearest down to, say, distance from the Sun, why is there such a disparity in temperature? and as a clue to answering that question, here's a nice sunny day on the moon that was actually this wasn't the slide I was going to show but anyway the last time I gave his talk was at Trinity College so I tried to find a sunny day in in Dublin this is a challenge so I didn't quite succeed You just cheated and put in the background, it's a blue sky. But anyway, that's for the sake of argument, here's a sunny day in Dublin. So the question is, what's the difference? Now, well, what's the difference? Strength of shadow. Right, so one's blue and one sky is blue and the sky is black. Now, the sky is blue because there's an atmosphere and photons are being scattered off to the air molecules. So the reason why it's warm is because there's an atmosphere and the other is cold, because there isn't an atmosphere.

27:30 Now that actually, in a way, doesn't say everything, because it turns out about 99%, I think, if I can remember my numbers, of the atmosphere is made up of nitrogen and oxygen. so those two elements if you like the molecules make up the vast bulk of the atmosphere and yet if those were the only two molecules that the atmosphere was made of there would actually be virtually no difference between the temperature of the earth and the temperature of the moon so actually 99% of the atmosphere is actually doing nothing to regulate the temperature of the earth what is going on. So here's the next little animation. So here's the Earth scene from actually space. This is a kind of a montage of images from satellites in geostationary orbit. And they show the Earth, or at least they show the atmosphere as a kind of complicated beast. You see weather systems moving from west to east, and there's something going on during the tropics. Now, what we're actually looking at here is the Earth in the infrared part of the spectrum. So these infrared radiometers on the satellites. And I remind the audience that infrared is measuring heat. So there's a cat in infrared. And also... All the opioids will apply. I'll show you what's happening. Yeah, so everything releases everything terrestrial is radiating heat and including the earth itself and so if you go back to this slide here which I made it once again what's actually interesting perhaps is that we're not seeing the surface of the earth we're not seeing the heat really coming from the surface of the earth we're seeing infrared radiation coming from something in the atmosphere because we've clearly seen these patterns being influenced by the weather. So there's some gas in the atmosphere

30:00 that is actually trapping the heat of the Earth and re-radiating it out at height at a colder temperature. So whatever this gas is, it's actually acting as a blanket, trapping some heat in. So I ask people what this gas is, The answer is... Carbon or out of that? No, it's not. I'm glad you said that. It's water vapor, actually. It's water vapor. Now, just to make absolutely sure, I mean, this is a sunny day here, but yet there is water vapor in the atmosphere. Water vapor is the gaseous form of water. So it's completely transparent to visible and indeed ultraviolet light. But it's opaque to infrared. And water vapor is actually the primary what's now called greenhouse gas it's it's the number one uh gas for uh for regulating maintaining the temperature of the earth this was actually realized by um an irish physicist um sort of john tyndall um who uh back in these you can see from the photograph victorian chat had a magnificently victorian turn of phrase too Brilliant. I just wish I could write like this. This aqueous vapour, water vapour, is a blanket more necessary to the vegetable life of England than clothing is to man. Remove for a single summer night the aqueous vapour from the air that overspreads this country and you would assuredly destroy every plant capable of being destroyed by freezing temperature. The warmth of our fields and gardens would pour itself unrequited into space and the sun would rise upon an island held fast in the iron grip of the cross. We would be on the right like that if we didn't have to learn cavities here. So, I just want to say a bit more about water. Water is a key quantity in the atmosphere. So, here's another one of these animations from space, from a geostation satellite. We're looking down over the sort of western, tropical western Pacific. This is China, coastal China. Here are two lovely, by the way, hurricanes or tropical typhoons, as they're called, typhoons as they're called in that part of the world. But I'm going to show this again because what I actually want to show you... So this is an area where there's thunderstorm clouds. Look at this.

32:30 Thunderstorm clouds being created. And what's happening is it's sucking up air into the atmosphere. into the high atmosphere and throwing out water in the high atmosphere. So these thunderstorm clouds, particularly in the tropics, are a source of water in the high atmosphere, and they're a source of this kind of greenhouse gas that's keeping us warm. That's a key point later on. Okay. But there is a second greenhouse gas. Molecule to molecule is actually not as important. But it has this trend. And this is a measuring station in Hawaii that's been measuring carbon dioxide concentration since about 1958. And we're going up pretty much the present day. And what you see is, unlike water vapor, it's got this inexorable upward trend over the years. It fluctuates like this because of the annual cycle. So in summer, the vegetation of the northern hemisphere is obviously more active, and that takes in carbon. And in winter, so in summer, there's an intake of carbon from the atmosphere into the biosphere, as it's called. And in winter, the biosphere is less active, so there's this annual cycle in carbon dioxide. But the soon problem goes on that is this trend. This is in parts chameleon. We're down here at about 315, I'll give you 380. Now, it is, I think, one of the uncontroversial, even the most diehard climate skeptic has to admit that this increase in carbon dioxide has been caused by man, caused by mankind burning fossil fuels, either through coal-fired power stations or jet aircraft or motor cars and such like that. exponential well it is what it is I mean yeah well it's up to us this by the way I was going out to London the other day and there's a newspaper so normally they're talking about

35:00 interest rates hitting 20 year highs but here CO2 levels hit 30 million year high. So this is on the basis that the values of 2005 have just been released by the people in Hawaii and it's 381 parts per million. Now I think it's 30 million years that's living in saturation but you can go back and look at ice cores and find little bubbles of air embedded in agreements in Antarctica and such like. So you can measure CO2 in air that's, you know, a million years old, at least, anyway, and certainly 381 parts per million is more than has ever been measured in these ancient air pockets. So we're talking about levels which are certainly unprecedented in the sort of, well, last million years, anyway. And so it's much of an interesting, from the ice core evidence, can one show a continuously in trend or are there drops and is there a periodicity there's very little actually it's fairly continuous actually um through the ice ages that it tends to go down it is true i mean during the glacial interlacials there are there are fluctuations um but um uh well i mean what i should say sorry i mean but yes so during the glacial interlacials there are fluctuations but Let me start my answer again. It's been increasing since the industrial revolution basically. Now since then it has gone up and down basically with ice ages. So if you go over a million years then you can look at... You mean before that? You mean before that? You said since then. Since the last ice age it's been fairly constant. Except since, you know, it's about 1800. When it's been rising clearly because of... So most of the variability is associated with ice age. Right, right. Now, this statement is really the crux of why there are uncertainties about global warming. If you remember, I said... I mean, there are a number of uncertains, but if I had to isolate one, it's this. I said to you that basically it's water vapor that is the key greenhouse gas. CO2 is a kind of secondary one.

37:30 CO2 is increasing because of man. The question really is, how will increases in CO2 change the water loading of the atmosphere? So naively, one would sort of say, and indeed many of the sort of simple models have this in it, that as you warm up the atmosphere, say if you start to warm the atmosphere by increasing CO2, then this evaporation from places like the tropical western Pacific will increase, so there'll be more, you know, a warmer atmosphere can hold more water, basically. So as it holds more water, then that increases further the water content. But the problem is, as I showed you, that water is a desperately, in reality, it's a kind of complicated gas. It's not well mixed. And that's one of the reasons why we use models. Just, sorry, before I come to talk about models, this is some historical evidence that we are seeing a problem. So the instrumental measurements of temperature around the globe really only go back to the late 19th century. I mean, places like Oxford have temperature records right to, I don't know, 1600 or something, but the trouble is that's just one location. If you want to get some measure of global temperature, then obviously the global records are, you know, it's just like 100 or so years. If we look at temperature over about the last 100 years, over land we see this over the oceans we see that and the combination we see that. So we are seeing an increase in temperature of, I don't know so are the land maybe on the path yeah this is Celsius this is an American thing, this is paragraph itself. So it's about one, say one, one of the big degrees over land and dip that's over the ocean by the way you'd expect smaller, the surface of the land has got a smaller heat capacity in the ocean and sort of transport heat down depth, so you tend to get less of a surface change over the ocean of land. Now, this itself doesn't prove anything because, of course, some people say this could just be natural variability, and I agree, this itself isn't a proof of man's impact

40:00 and climate. This is an attempt to use things like tree rings and ice core data and stuff like that to kind of go back to an earlier epoch. So this is an attempt to reconstruct global temperatures to about 1000 AD. This has generated a lot of controversy because climates, people that don't like global warming, talk about all the uncertainties in this type And these calculations have not been done by me, and I'm just showing them this is what seems to be suggested if one tries to reconstruct temperatures back a thousand years, that this period of warming that we've seen in the last few decades seems to be unprecedented. but that's a difficult one. So I want to come to an area this is where I work in which is the whole area of modelling. So I'm going to show you another animation of the world but now not from a satellite, from a computer. So this is looking inside a computer model. And this is again showing, actually showing precipitations of liquid water if you like. And again, you can see the sort of complicated weather patterns are being generated fairly sort of realistically. As I say, the reason, yeah, so just to say once more, the reason we use these types of models is that to understand climate change, this interaction between CO2. CO2 is a pretty well-mixed gas, so the 381 parts per million don't vary. If you measured CO2 around the world, you wouldn't see big variations in CO2 concentration. It's very well-mixed. But water, water vapor and water content are highly inhomogeneous. And so to try to really understand properly the interaction between increasing levels of CO2 increasing levels of water we need these kind of realistic models yeah how what are these models are these sort of up in each model so I'm going to talk about that that's just coming up oh here they are yeah so so here's my so yeah people think these models

42:30 are just you know my prejudice about what this is what do the public think or they're just a and glorified Sony walkstation game, etc. Well, no, they're not. They're based on this guy here. Now, he didn't know anything about global warming, but his physics is put into these models. This is a slide which I just showed. Now, the public don't understand this equation, of course, but I point out, by the way, I showed this because... This is Stokes, I gave this talk in Ireland, Stokes is in Ireland, I've yet to be invited to France to give a talk, if I was I would have a picture of Nunny in Ireland, and silly yourself invited. I just explained, I mean this is basically an expression of Newton's law, I mean I talk about Newton's law, I don't know if it's an English order and something like hitting a cricket ball with a cricket bat or something and the fact that you hit it with a force and off the accelerates. So this is the same law that applies to the whole atmosphere. Now I try to liken this to I say this is a fantastically beautiful equation, which I believe it is, and I liken it to a work of art. We had a talk about the connection between science and art, so this in my view is a work of art. But the piece, but rather than sort of liken this to a Renoir or a Matisse something like that. I liken it to this Russian doll here. And the reason I liken it to a Russian doll is that this, by the way, this is a rather special Russian doll because it unpacks, not into just a few Russian dolls, but even more Russian dolls and ends unpacking into yet more and yet more still and yet more still and so on and so on. And the reason The reason for saying that is, as you all know, that this is a partial differential equation and if you actually want to solve it on your computer, you have to first of all do a similar unpacking, if you like, into ordinary differential equations. And the problem is that this is beautifully elegant, and the beauty kind of delies its complications, because it describes scales of motion from the very largest scales on the planet, tens of, you know, sort of jet stream meanders of wavelengths of tens of thousands of kilometres,

45:00 right down to little, you know, sub-millimeter scales. so this Russian doll the biggest Russian doll of my Navier-Stokes equations has about a 10,000 kilometer let's say height the smallest is less than a millimeter and moreover these equations are non-linear and what that means is what I say, what it means is if you jiggle this big guy this big doll, I mean it will jiggle all the other ones down in turn If you jiggle the small ones, they'll jiggle the back ones, the big ones. So you have this sort of energy being propagating backwards and forwards across scales, which means you can't, you know, you can't just look at this one or this one or this one in isolation to get the whole thing together. And herein lies, you know, the problem that even the world's biggest computers aren't big enough to represent all those scales from tens of thousands of kilometers down to little ones of less than a milliliter. So what I have to do is truncate. I mean, I definitely want, I'm interested in these, I definitely am interested in the very big, I want to know what global warming is, and I want to know what, this one could be the size of North America or Europe, you know, this one could be. How do you find the scales? How do you find the scales? Well, these are done by, actually, yeah, I'm all familiar with spherical harmonics. what we actually do is decompose the um actually come back to this equation yeah we actually project this onto a spherical harmonic basis and so the differential equations are the differential equations for the coefficients of the spherical harmonics there may be hundreds of such there are hundreds of such but actually this is taking place on the sphere this is taking place on the sphere yeah i'm sorry this is all global these are global models so so the number squared is is still a plus in all the spheres. Exactly, that's right. And we have to do it at each level as well, so there are potentially levels going from the ground up into the metasphere. So actually, we end up with... we typically end up with 10 million in these models. So we can't, even the biggest computers can't represent everything down to sub-millimeter scale. So we actually have to truncate. So we get rid of a load of these, and we get rid of some more, and get rid of some more,

47:30 and get rid of some more. And now we've got, let's say, down to about 100 kilometer. So that would be a typical minimum scale in today's global climate. And typically, bearing in mind, I showed the momentum equation. we'll also have our temperature and water and maybe other constituents that we're interested in in the atmosphere. So typically we have about 10 million ordinary differential equations that are actually in these global climate models coupled non-linear differential equations. But the problem is that you can't really neglect and together the stuff that you have decided you haven't got a big enough to represent explicitly, so we have to do something, and I represented what we do by this rather inelegant tin can, so this is not a beautifully handcrafted Russian doll, it's just a lovely old tin can, and this ugly old tin can represents what I'm calling simplified approximate formulae, I mean the word, it was often used, parametrization, They're parameterizations to describe the bulk effect of motions, of scales of motion that you can't resolve. Now, clouds, you see, are a part of that. I mean, individual clouds have scales of maybe a few hundred meters up to the biggest thunderstorms and a few tens of kilometers. So what we try to do is apply, if you like, the ideas in statistical mechanics to come up with kind of simple, if you like, you know, diffusive type bulk formula, which might describe the net effect of all these motions that you can't resolve properly. The problem actually is that the methods of statistical mechanics don't really work, because you don't have a scale separation between the things that you are resolving explicitly and the things that you chromatized. In statistical mechanics, you're only interested in macroscopic phenomenon at the microscope, whereas we're trying to apply the same techniques to kind of get simple formulae for things that we can't resolve explicitly, but they're not scales separated from the, so this is the real, one of the big issues in scientific uncertainties, that

50:00 because they're limited by computers and because a lot of the important things that determine climate, the water, as I say, the water, you know, the water, how water is sucked up from the ocean, how it's transported around the atmosphere. It's partly by large-scale weather systems, but it's also by small-scale things like thunderstorms. In fact, we can see, you know, we can see this at work, this problem. If we go to one of these climate models, and so you can see it's more like a kind of quantilist picture than the old satellite map and that's because we are getting down and seeing the granularity of the spherical harmonic truncations the models actually do produce have a kind of shot of producing things like typhoons but you see the representations of these clouds and things is not nearly as precise as in the real world the limitations of the computer basically um six o'clock let me just do you have five yes please i'm not going to get to the end of this talk i'll just keep the last bit uh for tomorrow but um to keep us alert this is so yeah this is the sort of thing you see uh so you can say are these models any good? And, you know, can I believe them for climate change? So I would say, yeah, they're very good because we test them, you know, every day in weather forecasts. And this is what my golfing friends say. Wait a minute. What about? And they always point to this star. I don't know. This is a guy called Michael Fish. Now, Paul Michael, well he's awesome. I mean, Michael Fish has made good living out of this forecast. It's his hurricane that destroyed my caravan. Oh, is it? A brand new caravan. I may have a picture of your caravan. Two weeks old it was my caravan. This guy destroyed me. In October 1987, he said, don't worry, interest, I should have a story, ask me afterwards why he said this, but anyway. Well, it's so good, you have to say that. He said, a lady in Bournemouth, was it? Rang me up this afternoon to say, there's a hurricane on the way. there's no hurricane. The more interesting question is that the lady was actually from France, and the question is

52:30 why did she know, or why did she think there was a hurricane on the way? That's the interesting question. He was telling me to ask you that. But anyway, he said on the news, don't worry, this lady stood out thinking there was a hurricane on the way, don't worry, don't be so stupid, England doesn't have hurricanes, there won't be a hurricane. And sure enough, you know, a few hours later, trees blown over, blown over. There's Basil's Caravan. And the town of Seven Oaks in Kent produced some no-oaks. There's Caravan. And there's Basil's Caravan. Almost all the trees. And just to give people who were, obviously, weren't in England at the time an impression, almost three-quarters of the trees in Windsor Great Park were uplifted and lower down. It was in... Which year, wasn't it? 1987, October 1987. Sorry, 30th, 30th. So this is actually a great example of chaos in action because this is actually a kind of retrospective forecast that we did, actually with our latest model, but going back to 1987 conditions. And what you're actually looking at here are... I can't actually remember now. is the, so these are surface pressure maps. So this was our best estimate of what the surface pressure, you know, what the weather, if you like, was like over the Atlantic. This was just a few days, a couple of days before the storm. And this is one where I have just essentially added flaps of butterfly's wings to the initial conditions. They, you know, by eye, you'd be hard pressed to see much difference. But they're very, very slightly different and if we let this thing go so we're going to animate this so this is the computer taking these initial conditions forward onto the morning of the storm now this is what this is what friend michael fish i mean it wasn't his fault because this is the guidance he was given basically by the Met Office. By the Met Office. I thought it was hardly born in 19th century. Trick one. Now, you know, while the forecasters are looking at this map, this is basically, this would be the morning, this is 66 hours, so this is, well, so 6 o'clock in the morning

55:00 of the, whether it is, 15th, 16th, 15th of October. This is actually a little ridge of high pressure, said is well after some little small disturbance we could clear away early in the morning we'll basically have a fine day however a little flap of a butterfly's wings to the initial condition and this is a very intense deep depression and if he had had that forecast um he would uh certainly have been able to uh warn you know people to This was actually a 66 hours. So when did you start to show it 66 hours? Yeah, I've shown it 66 hours into the forecast. So this is actually a nice example of chaos in action. So this is Lorentz, and he shows the same thing. component model, not a 10 million component, which is the way the one is. But you see the same thing. You start with virtually identical initial conditions and you develop a completely uncorrelated time series after a certain time. This is stuff you all know. So, oh, except you don't know this. Lorenz never really wrote about butterflies clapping their wings. He talked about seagulls clapping their wings and changing the course of the weather. So it's a bit like Macbeth, you know, he never said, well, he never said, well, he never So what we do these days, because, you know, in 2006, it's still the same, you know, we're still facing the same problem. What we can do these days is actually, you know, and the forecasts are these days are based on this type of approach. We don't just run a single forecast. We run the model many, many times here, 50 times, with very slightly perturbed initial conditions. And so this is a rerun of that October 87 case. And what you see is, you know, the number of the individual forecasts have developed these very strong storms, other ones don't have it at all so a forecaster, modern day Michael Fish would actually be able to first of all see that this was extremely unpredictable I mean this is an extremely

57:30 normally weather is much more predictable at this range so this is we're actually on a part of the attractor near where all those kind of instabilities are, it's where the locally eponome of experiments are just exploding and so that's the first thing he would see, that it's actually very difficult to make any sort of definite prediction. However, he would be able to at least alert people of a risk or a probability. And in fact, you can even see what we do now. We put this all into a probability map. And this shows the probability of a... Just where that pointer is, is my camera. Yeah, well, this is precisely the point now. You know, what he would say is the probability of both is about 40%. Now, it's still less than likely, if you like, a priori, it's less than likely that it will happen. But it's up to you, Basil, to make that, and this is what, this is the kind of theory. It's up to you to decide, do you want to kind of tie your caravan down or whatever? If you don't evaluate it very much. Move it from under the tree. Yeah. But it's what, you know, this is where this, this is where the notion of probability by the economic loss to you. So if the caravan is only worth £5, then you may need an 80-10 probability before you tie it down. But if it's worth, you know, £50,000 then maybe a... And I don't have that sort of car. Now, chaos plays a role in climate. And I'm going to stop, actually, in two seconds. But this is my model of climate. So this is a kind of a mechanical analogue. It's not exactly chaotic because it eventually stops, but whilst it's oscillating around these magnets, it's kind of, the time series of magnets is sensitive to the ways, how you start the pendulum. So I'm thinking of these as weather states. Now, when, so, you know, doing a long-range prediction states, it stops there, so it's not precisely held it. You can't predict in detail over the long period which Magnus is going to. But you can say the probability of it going

1:00:00 into any wonderness is 25%, so that at any one time, if you forget about the transition times, it's got a 25% chance of being under everyone to Magnus. Now, climate change, and is like sticking a little wedge underneath this magnet. So the thing is still chaotic. And if you think of this as a cold winter, every now and again, who's asking me about the winter? Somebody asked me, oh, yeah, Maurice, yeah, the cold winter we had. Well, you see, you know, in such a chaotic system, yeah, these things happen, occasionally you get cold winters. but you're biasing now the probability of occurrence of this guy is less than this guy and the challenge in climate change yeah, there we are, we had a cold winter but that was just one of these chaotic fluctuations which are becoming increasingly less likely so what I want to do in the last bit of the talk and I'm going to stop now is actually show you some predictions of climate change. And they're going to be based on this idea, first of all, that climate is a chaotic system, to compute these changes in the probability of a current of different types of weather. We're going to run computer models many, many times. But what we're going to do is vary not only the initial conditions which are uncertain, but even more importantly, these uncertain bulk formulae, because this is how, the fact that we can't use Navier Stokes right down to cloud scales because the computers aren't big enough is one of the key uncertainties. And what we do is vary the parameters in these uncertain bulk formulae within these ensembles. So I'm going to show you results from running many climate models with slightly different bulk formulae many times. And I'll show you basically this will give us kind of probabilistic forecasts of how temperature will change in the next hundred years, but even more important perhaps how precipitation will change. Because things like droughts and floods, particularly in parts of the tropics, are going to be one of the key things for society, and estimates of how extreme changes in climate. So, yeah, so the next few slides are actually about results, but I'll leave that thing for tomorrow. Thank you.

1:02:30 But don't forget the first bit of the talk, because I came here. So, I think it's time now to move that up, right? Oh, sorry for the camera. Yeah, I was going to say, let's...