 Okay. All right. Well, good morning, everyone. Today, for those of you who are not yet awake, it's all right. We're talking about Dream here, the model. Actually, after my experience yesterday with the Speedy model, I decided to change the title of my talk. All right? So this is a new title. Although slow and steady wins the race is what they say. Now, I can crash my model as well, though. So this is an empirical, dynamical approach to teleconnection. So little by little, I will explain what that means. So let me just give you an outline of the talk to start with. So this is the Dream model. I've had this model for a long time, and I've only recently decided to give it a name. And it's called the Dynamical Research Empirical Atmospheric Model. So empirical is the key word here. It means that we're actually appealing to data to provide some of the forcing functions in this model. And most of the work that I'm going to show you is with my long-term friend and collaborator, Stephanie LaRue. And recent stuff I've been doing in Brazil with Tosio and Jose Leandro in Sao Paulo. So imagine that I'm the director of a large climate research center, and I'm giving you an overview of our activities. What normally happens is that the first slide looks like this, and it's a very nice nature. It's beautiful. We have trees. We have clouds. We have volcanoes. And in there somewhere, look, there's that arrow there, wind. See? That's atmospheric dynamics. And you think, well, this is inspiring. And the first slide of these talks always looks like this. Second slide of the talk always looks like this. And this is the point. So you have all sorts of where you have your core, your dynamical core, your atmosphere and your ocean. And then you have all sorts of other stuff going on. And this is necessary for a comprehensive simulation. And yet it can interfere with understanding the contribution of the dynamics. So there's different approaches towards stripping this away and trying to concentrate on the dynamics. And the speedy model is a much leaner beast than this, but it still has a fully physically based approach. What I'm going to do is go one step further and just remove everything and replace it with data. That's our approach. It goes with salt. So this is the dynamical research empirical atmospheric model. There's a link here, but essentially it's a link to a presentation which is very much like this one. So for you, no much point. So I'm going to talk about the way in which this works to start with. Then I'm going to go into some detail about a new piece of work on adding an annual cycle to this model. Then I'll talk about some perturbation runs we've done for remote influences on South American rainfall. A little short section on condensation heating which I've put into the model. And then if time allows, various other bits and pieces which illustrate the types of things you could do with this modeling system. So I'm going to flip through the first bit very quickly because it's not ready for this presentation, but I mean it's a primitive equation model. You can imagine the momentum equations like this in a very simple form. You can form a vorticity equation like this. The real momentum equations, the primitive equations have more terms. They have these advection terms. They have Coriolis term. But you can actually still write them in this way and that's what Hoskins and Simmons did. So this is from their 1975 paper. And you can also write the divergent equation using similar terms. So the Hoskins and Simmons primitive equation model, and this is a scan from their paper, solves for the vorticity, the divergence, temperature, surface pressure and there's the hydrostatic equation. So five equations and these are the variables which the model solves for vorticity, divergence, temperature, surface pressure and these are some of the names which are given to some of the other terms in the code. This is supposed to be for a very hands on type training course. The time scheme is a semi-implicit scheme so you can summarize the divergent part of the flow with these equations and you can eliminate for the divergence and you get this elliptic operator, the wave operator for the gravity wave equation which has a source here. So we have fluxes, sources and the gravity wave source and it's that part which needs to be solved semi-implicitly. That means that the time step has a kind of average between the present step and the future step to filter the gravity waves and allow a longer time step. But I'm going through this very quickly because it's not our main interest today. It's a spectral model. It is a triangular truncation so this is a zone wave number, a meridional wave number and it's a kind of funky jagged triangular truncation which means that for each zone wave number you have the same number of even about the equator and odd about the equator spectral coefficients. Traditionally that was useful for doing hemispheric runs. So I said I stripped away all the non-dynamical processes. It's not quite true. I have some dissipation in the model and this is something I've kind of tinkered with and tuned. So it has some scale selective hyper diffusion and it has a restoration throughout the troposphere towards, well, a restoration on temperature with a time scale of 12 days independent of height. That is, so that's just a Newtonian relaxation and on momentum and temperature there's also vertical diffusion which is max in the low levels, a bit less than a day at the time scale and then also present in the free troposphere but much weaker. So that's about it really. There's a bit of extra drag over land but that's about it for the physics in this model. It's just a bit of that thing. So the key is the foresail. Before we get onto that, I'm just going to skip that but that's technical. It's a sort of community resource, this model. You have to join the club. It's a GitHub page. And the dataset. That's the all important thing here is the dataset. So we use, recently we've switched to ERA interim. So we have 38 years of data for the model variables, autistic divergence, temperature and now specific humidity as well I've included in the model. There's no orography in the model so the surface pressure is calculated using a hypsometric equation and the mean orography, the effect of orography is represented by the effect of the foresail, which I'm going to calculate. The timing is we have 38 years of data. So from 1979 to the end of 2016, the data is 55,520 records, four times daily data. So this is a very useful table for just keeping track of how the model accounts for time. And an annual cycle of 365 and a quarter days gives you 1,461 records. So onto the core of ERA interim. How this works. So I'm going to slow down a bit now. So just for the purposes of illustration imagine you have an equation for a tracer cube. So which is advected and there's a forcing and there's some linear function of q which is the dissipation. If you separate that into a time mean component and a transient component, then you can write this development equation like this. The advection of the time mean of q by the time mean flow will be balanced by the damping on the time mean, the time mean of the forcing and this term which is the time mean transient eddy fluxes. This is a kind of non-diversion expression but it doesn't make any difference. The time mean eddy fluxes can, this is dead, the time mean eddy fluxes can be interpreted as a component of the forcing. So that's a kind of intuitively easy to understand way of looking at it. I'm going to reinterpret that in a more generalized way now just by thinking of phi as the model state vector. So phi, well, it's the state vector of the observations. In fact, it's everything. It's u, v, q. And so this will develop according to some non-linear advection operator, some linear dissipation operator and a forcing. So that likewise can be separated into time mean and transience. So in this then development of this phi dash is this operator applied to the climatology and the perturbation and there's a mean forcing and a perturbation forcing which forcing is in general a function of time, okay? Or you can write down the equation for the perturbation. You can write it, well, it's the same equation. You can write it like this. There's a time mean component. Oh, thank you. There's a time mean, the action of the model operator on the time mean. There's a linear operator which is the linearization of the model about that time mean and there's a second-order term. And if you then take the average of that equation, the time mean budget is, again, it's equivalent to this. There's the action of that operator on the climatology. There's a second-order term which is the transient eddy fluxes and there's the mean of the forcing. And then if you look at, if you subtract that from this, you get the eddy budget equation and that is you can develop a perturbation according to a linear process around that climatology and a second-order term. Now the advantage, this is to illustrate the different types of separation you can do because this is my advocacy of using the time mean as a reference because the advantage of that is that this equation, all the terms in this equation, average to zero and they can be, since they are variations around the observed time mean, they are actually realistic. They look like atmospheric variability. If you wanted to be a bit more purist about this, you would say, well, that's no good to use the time mean because it's not the solution of the primitive equations. Use something which is a solution to primitive equations, then you'll have a better posed linear problem. But your perturbation will not be small. It'll be something large and something that doesn't look like anything like atmospheric variability. So the relevance of your linear problem is questionable. So this is the trade-off. So yeah, that's just a little bit of philosophy. How does that relate to actually doing something practical like forcing this model? So let's go back to the development equation and I'm going to slow down even more now. So this is phi, the observed state vector. This is the model of which an advection term which is nonlinear and a dissipation which could probably be linear. And a forcing which is unknown and a function of time. We don't know what that is. The task now is to find it. Now, if we introduce a model and we call it psi just to avoid confusion with the observations, then this is the model operator so we know what it is. And what we're looking for is a forcing, G, which I will assume and this is a big assumption that G is independent of time. So I just want some practical model where I can just put something on the right-hand side of the equations on it and it will behave like a GCM. So I'm saying that G is equal to F bar basically. It's as simple as that. G is equal to F bar. How do I find G? Well, we can use the model and the data. So imagine that we run the model without any forcing. So just put zero on the right-hand side. Then just run it for one time step from some initial condition. Then the tendency that the model gives you will tell you what this is, this operator given that initial condition. So now let's just say I use the observations as an initial condition and I have a whole string of observations of 55,000 realizations of the observed atmosphere. I'm going to use all of them and I'm going to do this 55,000 times. Just one time step. And G is just the average of all those. If I say G is equal to the average of all these one time step forecasts, it will be the average of A plus D acting on phi, which is F bar. So that gives me G equals F bar. I can use the model on the observed data to find the forcing. So then I take that forcing that I've diagnosed and put it on the right-hand side and I've got a GCM. So that's the trick. The first time you encounter this it takes a little bit of time to adjust to. So I'm just going to stop for 30 seconds there or 10 seconds. It can be, but it doesn't really matter. You just get a bunch of observations and I indexes the observations. It doesn't necessarily have to be in sequence. You just have to have a representative average set of atmospheric states. I will get to that. It is equivalent, but it needs to be explained why it's equivalent. I've got a slide about that. One thing you might think is maybe this is cheating. Maybe we're just putting the answer in. Of course we're just putting observations in. That's not a model, that's just cheating. Well, no, it's not cheating because it's formally what the model guarantees is that this is true. So the average, the time average of A plus D on the model is equal to the time average of A plus D on the observations. That does not mean that my model climatology phi bar will look like the average of phi. Psi bar, sorry, will look like the average of phi because you break this down and it has two terms. It has the time mean term and the transient term and they could balance differently in the model than in the observations. So the model might have systematic errors in its climatology and it will also have compensating systematic errors in its transients. And so it depends on whether the model is solving the right problem, whether or not it gives a good result. So it is actually a proper model with its own internal variability and everything and we'll see how good the result is. So this is not the first time this has been done. I didn't invent this technique. The first person to do it was John Rhodes and then it was done by Marshall and Maltini and then Fabio D'Andrea had a crack at it and then Hylin and Jack Durum were doing it when I arrived in McGill and I think the first person to do it was the primitive equations and then there's a whole string of people who did it after that. So that's not all. I mean, it's nice to have a simple GCM but there's other things you can do. So let's go back to this equation again and wading through some technical stuff here. How about instead of defining a forcing G equals F bar, let's just define a forcing H which is equal to A plus D acting on the climatology. So it's not the average of A plus D acting on the data, it is A plus D acting on the average of the data. Now what happens if I force a model with that and if I initialize with the climatology? Well, nothing of course because H is equal to this and if my initial condition is that then there will be no development. H will directly cancel the model acting on its initial condition. So I'll have no development. I'll have a basic state which is my climatology or it could be anything you want and I calculated exactly what's necessary to stop the model from developing and this is what Jyn and Hoskins did and many other people, I think you cited Adrian Matthews did the same thing. They just find that and then run the model and it gives you a perturbation model because you've nailed the basic state in place so it's not like a simple GCM, it doesn't have a turbulent any field, it has a basic state and then you can introduce a perturbation in the initial condition and it will give you a linear model. That term will be small so you've got a linear perturbation model where this is linearized about your climatology in this case. So you can find solutions to that linear problem and I've done this, you can find the normal modes of the system you can find the eigen modes which have a structure and a frequency and a growth rate sigma and sigma, if sigma is positive then it's unstable you can find the unstable modes. Alternatively you don't put the perturbation in the initial condition, you can actually add a small perturbation forcing and this approach gives you these kinds of solutions where you have developing well this is the rospy wave that comes out of the tropical heating for example you can just nail it, bang in a tropical heating here and watch the wave develop and it will be linear provided this is small you can even imagine finding the asymptotic solution which is independent of time, it will provided all these modes are stable so all these values of sigma must be negative it will asymptote towards a time independent solution trouble is basic states tend to be unstable so you have to find a way around that. I'm getting ahead of myself so let's just talk a bit about the difference between G and H. We call our tracer advection formulation. So the time mean of this equation is the time mean advection of Q is balanced by the dissipation applied to the time mean of Q, a mean forcing and a transient force. So this is my time mean advection dissipation, this is my mean forcing which is G and this is my transient advection forcing and the sum of the two is H. So H is equal to the what you might call the diabetic forcing plus the transient advection forcing that's just another way of interpreting it a word on damping and restoration there is like the held Suarez approach where you have a standard radiative equilibrium state and you relax towards it and that's a way of forcing the GCM. In what way is my approach equivalent to that and you can just look at the equation. So I have a tendency a nonlinear term of forcing and adapting this is held Suarez, this is a tendency a nonlinear term a restoration time scale if you like, well one over time scale, a radiative convective equilibrium state and the model state where you restore the difference between the two. Which can be written like this, it's R times phi star which is G and R times phi which is this. So it's the same. Provided D is diagonal, it's the same. But my forcing is the restoration state divided by the time scale of the damping and my damping is just one over the time scale of their damping. The difference is that the restoration approach you specify some idealised radiative convective state which you've dreamed of. I'm more objective, I don't care what that state looks like. I could calculate it but I don't care what it is because I've found out the forcing and the damping I specify myself. So we're converging towards the same thing. So what does it look like? This is the forcing on temperature specific humidity and zonal wind at low levels, 80.85 and upper levels 0.25 and a lot of the forcing, so this is G a lot of it looks for winter season. A lot of it the strongest part of it is just putting in what the lack of orography is living out of the model. In Greenland and Antarctica always show up when orography is important and the Andes as well. The humidity is interesting, I mean what it's doing is the atmosphere brings in humidity towards the equatorial region and then throws it up into the upper atmosphere and it's removed by precipitation and it's resupplied by evaporation. So that's what my forcing needs to do. Vertical profiles of this forcing, well here we have G here we have H and the difference between G and H as I said is the chance in any forcing for temperature. So a lot of what it's doing is just redressing what the diffusion has done at low levels. So if you show it without diffusion it becomes very, there's not much forcing at all really in the temperature for what is this? Specific humidity it just shows you what I showed you before. You're evaporating it in the subtropics and raining it out at upper levels at the equator. The transients are doing something there as well. And then from momentum there's some orographic drag which you have to represent and the transients are doing what they do. The transients are accelerating the jets at the upper level. So how about let's get to the validation then. So how does it work? This is DJF and this is a perpetual DJF model experiment. So this is low level zone of wind really only wind and temperature. So this is a long term mean. So I've got my jets in the right places a bit weak in the sun hemisphere. I've got systematic errors. Temperature is very good indeed obviously that is largely a problem. And really on a wind the stationary wave structure looks good. Upper level winds again it looks very good. So don't have much to criticize there. Humidity well we do have a humidity variable in the model and we are forcing it and it gives you a pretty decent humidity distribution. We have sources and sinks of humidity that represent out evaporation and precipitation. Transients so these are the unfiltered transients for winter and you can see that it kind of getting the storm tracks in the right places it is a bit weak in the sun hemisphere. This is what is this temperature flux. Yeah sorry I should have said this is low level 850 millibar V-T-dash so it's really on a heat flux a temperature flux and this is the V-Q-dash so this is moisture flux. So we've got the storm tracks in the right places this is unfiltered remember. It seems to be a bit weak in the sun hemisphere in the northern winter and it is very weak on eddy kinetic energy especially in the sun hemisphere. This is the momentum flux so you can see that convergent momentum flux at the storm exit there. It's getting that. This is eddy kinetic energy it's getting it well located but weak. And so for the filtered transients so this is less than 10 day filtered some very nice storm cracks. For the temperature flux still a bit weak in the sun hemisphere for the humidity flux very nice and the momentum flux and eddy kinetic energy. So we have a GCM which we can use for all sorts of things. Why is this so? Do they tell you something? Do they tell you something about the nature? I always wonder why it's not quite performing in the southern hemisphere. And I think the model is free to choose whatever dynamical balance it wants and so the ferrule cell at upper levels has a balance between transient eddies and the Coriolis force and at lower levels it has a balance between the Coriolis force and drag. So maybe there's something wrong with the drag which is giving you a weak ferrule cell which is consistent with weak transients. But I don't know what to do about that. So yeah you can kind of look at balances and say what's different than the model to reality. The model is free to find its own equilibrium between the transient part and the mean advection part. I'm going to move on now to adding an annual cycle to this model. Are there any more questions before I do that? Yeah I just used DJF. A quarter of that 55,000 it helps you to use computer time. I have recently upgraded the resolution. It was T31L10 for a long time. Now it's T42L15. And I don't see much evidence that it's improved the model to be honest. Yeah. Alright so this is a kind of mini talk within the talk here which I've very recently produced so I'm still getting to grips with this. Everything you never wanted to know about the annual cycle but you were too polite to leave. This is what I did in Brazil this year. So well I'm going to skip through this fairly quickly but the idealized, the kind of understanding we have of the annual cycle is that it's something very regular and it just comes from orbital considerations and the sea surface temperature so you can represent it as a time development, some damping and some forcing which is cyclic. And the solution to this equation is an exponential decay and two terms one of which is in phase and the other which is in quadrature. The importance of these two terms depends on how much damping you have. If you have a weak dissipation it's out of phase so that's like the ocean response and if you have strong dissipation it's in phase. That's more like a balanced atmospheric response. You can subsume this into the forcing if you want and just talk about the atmosphere and it gets back to some sort of restoration thing. But this is all kind of linear and assuming that the model responds linearly to a regular forcing and the model is non-linear so if you add a perturbation and add a non-linear term then you get all sorts of extra terms in the forcing and then the question is how important are those extra terms? There are interactions between, so if the twiddle is the annual cycle and the dash is everything else, there are interactions between the annual cycle and the mean, between the annual cycle and itself between the annual cycle and the transience to look at. So suddenly you've got a lot of terms to consider in the forcing. So we're going to explore that a little bit. Here's just a GCM experiment by PSUTI and Batiste where they looked at the importance of the SST for forcing annual cycle and the radiative forcing and they find that both are important. So what do I do? I've got to go back to my forcing strategy and now I have to think of F as having an annual cycle in it. So this is my model, these are the observations but this is the model and now I want G not just to be F bar but F bar plus F tilde. So tilde is an average annual cycle. So we cut up the flow now into three components, the mean annual cycle and the rest, everything else. But G, I don't want transience in G, I just want G to be a steady cyclic forcing. So anything which is not either steady or cyclic is because of the model's own dynamics, it's not coming from the forcing. So you can actually write down an equation for this F bar plus F tilde in terms of the model operator, assuming D is linear still. So this is like, you get a lot of interaction terms between non-linear terms involving these three different components. So you have the mean-mean and you have the advection terms between the mean and the annual cycle between the annual cycle and itself and then you have the interactions with the transience. And these all have aspects, these all, well, some of them contribute to the time mean, some of them contribute to the annual cycle. So I've broken it down into, oh and yes, there's also now a tendency term which has nothing to do with the model. You have to calculate that straight from the data. So I just do the center difference to find that. So I have my mean-mean term, my mean-cycle term, my cycle-cycle term, my cycle-transient term and my transient-transient term. And we can calculate each one of those components of the forcing and look at them individually. So isn't that fun? You can have so much fun doing this. So the question is how do we find them and how do we separate them now? So this is a bit fiendish. Well, first of all the tendency term, I've killed the or your battery again, that's fine. The tendency term is calculated straight from the data set. The mean-mean term is just like before, it's like h, it's like finding h. So you just have the initialized from the climatology one time step and you've got it. The mean cycle and cycle-cycle terms, well, that is like initializing from one average annual cycle. So 1,461 single-tie step runs is all you need. And you've got mm plus mc plus cc, which is from this. You already know mm, so you've got mc plus cc. How do you separate the two? Well, you do it again, but you put an arbitrary factor in front of the annual cycle. And then you use a bit of algebra from those two runs to separate them. And you've got mc and cc. So then you're ready to try and find transient terms. So you start with the whole data set, 55,000 runs from 5 bar, 5 tilde, and 5 dash. That gives you mm plus mc plus cc plus ct plus tt. You know these three, so it gives you ct plus tt. How do you separate those two? Well, you put an arbitrary factor in front of the 5 dash, a bit of algebra, and you've got those two, so you've collected them all. So what are these terms? What do they contribute to? The tendency just contributes to the annual cycle, not to the mean. It's linear. Mm only contributes to the mean and not to the cycle, but it has linear and nonlinear components. mc does not contribute to the mean, only to the cycle. It's linear and nonlinear. These other terms are just nonlinear and they contribute to both cc, ct, and tt. So let's have a look at them. And again, the fact that I have this vertical diffusion in the boundary layer kind of interferes a lot with the solution, so I'm just going to remove the vertical diffusion in the boundary layer to clarify things. This is using the model to do diagnostics on the interim data set, basically, in a kind of novel way. So this model is not just for doing simulations. It's a diagnostic tool. So this is my implied forcing, mm. This is my transient adduction term, and this is my total forcing. So you can see it shows up right before for the humidity and for the winds. And we can break it down season by season. And then we start to see the contribution of the tendency term. And it's extremely small. So which terms are important? Well, the mm is obviously important but it has no seasonal cycle. mc is clearly very important. This is temperature and humidity forcing. Cc is not negligible. That's the cycle interacting with itself. So you see it's like the combination of the time mean flow advecting the annual cycle of the tracer plus the annual cycle of the flow advecting the time mean of the tracer. Then we have tt, which is as before as the transient eddy fluxes, excluding the annual cycle. And that's important just as you'd expect it to be. Well, ct is rather unimportant. There's a big separation in time scales between typical dynamical transients in the atmosphere and the annual cycle. So they don't interact very much. So we'll just skip through the seasons. You get the same story every time. And I just want to concentrate in on one thing, which I found to be the most interesting aspect of this, is the maintenance of tropical humidity. And so if you look at mm, this is the annual mean of these forcing components. mm basically has to provide rainfall in the tropics and evaporation in the subtropics. You see the ITCZ in there. CC, the cycle cycle term that I just evoked, seems to have a clear annual mean signal over West Africa. So we're multiplying something over West Africa with the cycle cycle interaction. ct is negligible and tt is basically extra tropical. So what's going on? Let's take it down into the four seasons. DJF, MAM, JJASON, and there are three plots for each season, MC, CC, and tt. Let's just look at MC to start with. You see that the Hadley cell is ascending to the south and descending in the north. That's the kind of DJF Hadley cell. It stays like that through MAM and then it flips in the northern summer and it's rising in the north and descending in the south. And then it stays like that into the autumn. Now if you look at the CC term you have something which is kind of rising in the north and descending in the south over the West Africa, over the land mass. And it stays the same in March, April, May and it stays the same in the opposite season. It doesn't change sign. And so the African monsoon is... exists in the summer because it's a reinforcement between the MC term and the CC term. And it's not there in the winter because there's a cancellation between these two terms. So I'm still getting to grips with this but my understanding of my results, I'm just going to read this to you. So the CC term over West Africa remains the same sign in opposite seasons leading to a partial cancellation of MC in DJF and reinforcement in JJA. We can explain this in terms of seasonal anomaly covariance. The flow reverses but crucially so do the seasonal anomaly humidity gradients. So the covariance between the divergence anomaly and the humidity anomaly retains the same sign leading to drying in the guinean zone and moistening of the Sahel in both summer and winter and all year round. So in the winter this partially cancels the linear component and in the summer it reinforces it. And so cyclic changes in wind direction shift the humidity distribution. This is the onset of the African monsoon and that then interacts with the seasonal anomaly flow and it's this covariant interaction that characterizes the African monsoon. As I say I'm still getting to grips with that. Let's just step back now and look at the validation of this model with an annual cycle. And so we have our two specifications for the forcing. We have the cyclic in red and the old perpetual version in purple and on the right we have perpetual runs for summer and winter and in the middle we have the full annual cycle run which I ran for about 12 years and then just looked at the seasons. And on the left we have the data and so you can see that the model is basically doing if you just look at the first and third rows you can see that the model is doing something very similar with the full annual cycle as it was with the perpetual simulations. And that is consistent with the fact that we have very small contribution from that tendency term in the annual cycle forcing. So that's the end of the section on the GCM. The moisture budget over West Africa is an interesting case of looking at the annual cycle budget. The budget separation highlights two phase onset, reversed winds followed by an interaction with the modified humidity gradient. The perpetual runs are consistent with the small tendency term and this technique that I invented to separate out all those terms can be applied to other types of time scale separation as well so that's something in perspective. Any questions at this point? How am I doing anyway? I've still got loads to talk about. How long have I got? Tell me honestly, how long have I got? Oh god. There's a huge advantage to having an annual cycle in a model and that is that it corresponds one to one with the data set. So if I want to nudge part of my solution I can just nudge it and it just tracks in tandem with the data set. There's nothing to worry about. It's very much like what I was showing yesterday with those boundary conditions. So for me that's one big advantage. The other again, I mean it's a diagnostic tool as well as a model. I'm going to run over time a little bit but I want to show you some perturbation runs because this is the kind of thing you might be able to do this week. So here we're going to get interested now in South American rainfall. So this is the stationary wave number which is kind of, it's a graphic which gives you an idea. It's like a refractive index for Rossby waves. They're attracted towards low values of this and they're not allowed to go into these white areas where we have easterlies and the wave number is not real. So there's no propagation in those areas but crucially you see there is propagation in this area and we're interested in the rainfall here over Sao Paulo. So what could influence it? That's the question we're asking. What kind of teleconnections could influence rainfall in that region? And they could be Rossby waves coming from the southern sphere or they could even come through here. So we are going to do some modeling studies to address that question. So this is now using the model with that perturbation configuration. So we're forcing it with H and we're putting in a convective heat source. Just by way of example, this is a convective heat source over the mid-Pacific and the clock we're looking at is vertical velocity. So blue is up, red is down and we're going to run the model for 15 days with that convective heating in place. So just looking at the vertical velocity. So you see that downwelling Kelvin wave going round the earth and at the same time at slightly longer times here you see this extra tropical response developing. It's unusual to look at it in terms of vertical velocity and it's still there. Why are we looking at vertical velocity? Because that's one thing which might be important for rainfall. In this case you can see that obviously over the heating it's going to go up. It turns out that over the south polar region it's going down. So you might expect that to lead to dry conditions in our target region of interest there. So another thing which might influence rainfall is the supply of moisture. So this model has a moisture variable. So we can look at the vertically integrated humidity flux convergence or divergence. So what we have is blue is convergent and orange is divergent and we can just show the same run in terms of that diagnostic and obviously it's converging at low levels where you're heating but it's more complicated because you've still seen the Calving Wave. It's interacting with the climatological humidity field. So it's the dynamical response interacting with the humidity field. And it seems that it's quite consistent that you'll get divergence of humidity in the target zone. So this is really a drying signal for South America for that particular source region. You can look at the two of them together. So contours are humidity flux convergence and shading is omega. So where you see these two diagnostics agreeing that's the strongest clue for some information about rainfall. Even though it's just a dynamical model, there's no rainfall scheme in this model. Essentially they're talking about the same thing. Not really. This time, look at this. This is a scatterplot of one against the other. So this is omega. So downward this is dry and upward this is wet and this is the humidity flux divergence. So positive is dry, negative is wet. So yeah, it lines up in that sense it's telling you the same thing. But look at how it develops in time. It starts off very nice over the heating zone. And then as the teleconnection spread out, they become more and more conflictual over the glow of these two different measures. And it fills out that those four quadrants, depending on and just label them with different latitudes. So the bolder points are the tropical points and the fainted points. So they're not always in agreement. So we're trying to pick places where they are. So that's just one forward experiment. So we thought, well, we want to know about everywhere which might influence South America. So we're going to do all these experiments. Every last one of them. 15 day runs for each point. We're going to heat over each point. And remember, the blue points are done with a deep convective profile and the red points with a shallow profile. So we're going to, I can't show you hundreds of videos because I have like minus 10 minutes left. But I could show you what we call an influence function. So what we will do is we will look at this area here. And we'll bang in a heating somewhere and we'll look at the response at this point. But we'll plot it over here where we put the heating. And so that gives you a map like this. So for example, if you heat here, then after 15 days you'll produce a vertical velocity downward response here. If you heat here, after 15 days you'll produce a vertical velocity upward response here. That's the, oh sorry, no, that's the humidity flux divergence. So in terms of moisture supply, you'll have a drying response if you heat here, a moistening response if you heat here. In terms of vertical velocity, you'll have a moistening response if you heat here. So here you have an agreement. Both measures will give you a moistening response in the target time after 15 days. Now you can run this out. You can plot functions like that for one day runs, two day runs, three day runs. So you can actually put a movie of this function on and watch it spread out from the source zone as the lead time increases. And you see a backwards propagating Rosby wave. And if you use your imagination, you can think of it going through the wave guide as well. But the influence is fairly weak far afield compared to the influence of actually just heating over the target zone of course. Even if you heat over the target zone, you produce a conflicting response in this case. Because the onshore convection becomes important. So just remember this point here. So this is the best way to get rainfall in south powers to heat over the target zone. Because that gives you a big blue blob and a big blue contour. But there's a possible teleconnection here. Even if you heat here, then you'll get the right moisture flux convergence response here. Let's look at... What I'm plotting here... If it's not very strong, it's not plotted. I don't plot the zero right. But yeah. That on equator example I showed, apparently is a weak response compared to what Jose Leandro decided to plot here. And here's the scatter plot. So this is now an influence function scatter plot as it develops in time. It's usually consistent. The big cross is if you heat in situ. So that's not consistent as I showed. Most source regions will give you a consistent picture between those two measures. And just a couple more examples. And I think I'll have to stop. So how about this hotspot here that I'd identified. Let's just run a forward run from there. So this is now a forward run. God, I have to press it again. And it throws off a gravity wave to start with and then you get this effective development. And you can see that there's a consistent response over the target region there if you heat on the black dot. And finally, there's an example with ray tracing. So this is Jose Leandro kind of like doing this. You can calculate from that stationary wave, Rossby wave equation you can calculate the place, speed and direction of the ray. And he's plotted that along with the vertical velocity on the stream function. You see those rays going out from this example point here. Which is rather pretty. Yeah, you have to pick your level. So I don't know what level you must have used some level. Right, I think I'd better stop there. I've got to, I mean, there's other stuff in the overheads. Can I say one more thing? Because I've got something about adding moisture to Kelvin waves which is relevant to what we were talking about yesterday with speedy. Let me just show you one picture. So I've added a moisture, I've added a large scale condensation scheme to the model. So you do one of these perturbation runs where you have a Kelvin wave and you can plot it with various different basic states. And this is a dry model and then this is for the large scale condensation scheme. And so the dry model always gives you this fast regular Kelvin wave. You put in the condensation and it goes to the power and it produces these kind of patterns. So that's vertical velocity. If you look at the velocity potential, it goes, so what I've done is increased the constant, the L, the specific latent heat just buried it between zero and its correct value. And if it's very close to zero you get this dry Kelvin wave and then it just morphs into this slower response if you have that large scale condensation fully in the model. And that's just large scale condensation. There's no convection scheme there. Right, I really am going to stop.