 So good morning, welcome back after the coffee break. My name is Franco Molteni, I work in the research and Copernicus Departments of the European Center for Medium-Range Weather Forkast. I have two lectures in this week. The first one is a joint lecture with David and it will cover topics which are both similar to what Angel has discussed in his previous lectures, but as David anticipated, more from a more sort of large scale and historical point of view. So again, the main purpose is to give some foundations to understand how we can use circulation regimes to understand or to interpret tropical-extra-tropical interactions. First of all, let's talk about this concept of circulation regimes because this word is used in slightly different flavors and it can sometimes be a bit confusing. The idea of circulation regimes actually was mainly developed theoretically in the late 70s and the 80s, 1980s. And it actually came from sort of a way of interpreting observations which have been occurring for a long time about the fact that there are some patterns in the atmospheric circulations which seem to be particularly persistent or recurrent. Recurrent means that they come back quite frequently. And to see some example, you can look at the three maps in this slide. These are all maps of five de means fields of 500 hectopascal geopotential height during the winter in the northern hemisphere. So for example, in this first map, you can see that the flow is as very strong waves, a very strong ridge over the Rocky Mountains, big trough over east of Canada, a blocking high, big trough over central Europe, and then again a sort of blocking high here north of the Caspian Sea. So overall a very wavy pattern. If you look at the second map, you can actually see that this wavy pattern is still present, perhaps with some changes. This high is actually now in a more easterly position here over the North Sea. So in fact, you see clearly the sort of typical dipole structure associated with European blocking. For the rest, the flowing in the Pacific remains quite similar. When we now move to the third map, again the dipole over Europe is present with the blocking, the trough over eastern Canada, but now on the flow is a much more zonal structure here in the Pacific and the strong ridge has actually disappeared. So from the way I talked about this map, and it's a bit of a trick here, I discovered this map as if they were a time sequence of consecutive maps, but actually these maps occur each one in a different winter. So what it shows is that if you compare these two flows that actually occurred one year after the other, you can see they are remarkably similar from a large scale point of view. And if you compare this with this one again, one year difference, and if you just now focus on the European and Atlantic sector, then again you see this blocking structure looks very much the same. So a number of people have actually looked at this phenomena from an observational point of view, for example, Jerome Namayas for the Pacific anomalies or Rex, you know, they look at the European blocking, and so they noticed this characteristic of some patterns to repeat themselves in time. And so in the late 70s and 80s there were a series of papers about large scale dynamics that tried to interpret and explain the recurrence and persistence of these patterns. Nowadays we mostly refer to these patterns as either circulation regimes or weather regimes. Actually there's actually a sequence of dynamical concepts which are strictly related. Now if we talk about weather regime, we usually mean a persistent or recurrent large scale atmospheric circulation pattern which is associated with specific weather conditions on the region. This is what Angel has shown. So you may have circulation pattern that determines, for example, the patterns of precipitation. And the same thing is for the North Atlantic oscillation. You know, the different phases move the precipitation from Northern Europe to Southern Europe or vice versa. Actually the weather regimes are an application to the atmosphere or a more general concept which is the flow regime. So you can have a lot of geophysical fluid dynamics systems even very simple ones. That actually have an attractor which is complex so there are parts of the phase space of these attractors that are more populated than others. So again some preferred states of the flow that are quite different from each other and each of them is quite persistent. There's a famous analog for example in rotating annulus with some bottom obstacle that can either be in a zonal flow or in a very wave flow and you will find this example in many free dynamics book. Even more generally is the concept of multiple equilibria and basically whenever you have a non-linear dynamical system for some values of the forcing parameters in this model you may actually have multiple stationary solutions. So there will be a number of configurations where the dynamical tendency will vanish, will be zero. So in some sense this is perhaps the simplest concept that's looking for a stationary solution or instantaneous equation. Here you look in general to basically parts of the attractors that are more populated and basically the weather regime is the application of this concept to the practical atmospheric flow including the effect of baroclinic transients. So in terms of the dynamics these papers were mainly describing regimes as quasi-stationary states of simple equations that were just described either by the barotropic vorticity equation or basically for an equation that described the evolution of quasi-geostropic potential vorticity. So in practice Q in this equation is the potential vorticity and basically this equation can be simply stated that the vorticity is just advected by the non-divergent flow and you can actually derive the non-divergent flow from the potential vorticity itself. So you have two variables here but actually they are linked one to the other by a linear relationship. So if you know Q you know the divergent wind and then you have some dissipation processes which are usually linear in many models. They have the effect of a relaxed potential vorticity towards some sort of equilibrium. For example you can imagine that in the absence of forcing for the waves the system will relax to a zonal configuration. So in a multiple-equilibrium approach you just look for steady states of the instantaneous flow as predicted the pointer is disappearing a bit so let's look here. So you can just set the time derivative in this equation to zero and then you can look for instantaneous stationary states of this equation. So this equation is nonlinear because you have the product of wind times the gradient of potential vorticity and so it may admit multiple steady states and one can look for them. But yeah in the concept of regimes we are actually thinking at least for the atmosphere and at least in these kind of papers they were mainly trying to explain persistent anomalies. So persistent on the time scale of maybe 10, 20 days even one month perhaps. And so if you do that you should not just take the instantaneous flow but average this equation over a typical time scale let's say 10 days. And if you do that you basically can rewrite this equation where you have some terms where you just replace the instantaneous flow with the time average flow. And then because of the nonlinearity you actually have an additional term which comes from the nonlinear interactions or all the variability which has basically a shorter period than the period on which you are actually averaging. So typically if you are averaging over 10 days our clinic waste with a period of a few days will basically contribute to this term which is sometimes we refer to transient forcing or at the forcing in in this paper. So these are some brief list is not exhaustive. But just for some of the papers which are being quoted more frequently about multiple and quasi stationary states in in simple models. Perhaps one that is usually quoted as a starting point from this investigation is the paper by Charlie and the war in 1979. Where the authors look for multiple equilibria. What you see here are actually the titles of all the papers at the time of the paper was about trying to find multiple equilibria in a simple barotropic model of a channel flow with bottom. And they point out that basically some of the stationary states corresponded to different density of the zonal flow and the amplitude of planetary waves. David was working with with Charlie at the time and in another very famous paper in 1980. They extended the analysis to about clinic to level procedures traffic model and they looked again for multiple equilibria of of this model. It was more complex and had more equilibrium and of course if you want to know more David is here and you can ask for more details. And China shook line more return to barotropic theory but they put many more degrees of freedom in the barotropic model. The channel they work paper was based just on a reduction to three degrees of freedom. The grand deal again using a barotropic model with many degrees of freedom and they actually in this case look at the complex tractor but with strong persistent states. And then the Italian school came with a different proposal for multiple equilibria based not so much on the interaction between the zonal flow and the waves but rather by some nonlinearity of the wave itself which allowed multiple equilibria. And this particular theory came as a response to some observational studies that Souterra was doing at the time started at ECMWF and then he continued at Yale. That actually he could not find any evidence of multiple equilibria in the zonal flow which would contradict the Charlie and the work theory while he could find evidence in the structure in the amplitude of the planetary waves. All these models were basically or graphically force models so there was simple models with some usually simple sinusoidal topography at the bottom that would provide a forcing for the planetary waves. But of course and these were all models of the northern extra tropical circulation mainly during wind. Where during winter topography is not the only forcing for planetary waves and other important forcing is the Lansing thermal contrast. And so there was another group of papers and the first one was the one by Michel and the Rome in 1983 so they did a similar study but in this case the forcing for the planetary waves was not an aerographic forcing but a thermal forcing you induced by the Lansing contrast. And I will return to this thermally equilibration theory later on. Now all in all these models again the nonlinearity was just basically due to nonlinear interactions between the large scale flow. It could be the waves and the zonal flow or the waves themselves but there was no specific role for the high frequency baroclinic waves. So the concept of weather regimes as it is interpreted now can actually be traced to other papers when the baroclinic high frequency waves were explicitly taken into account. And the first one is the one by Reinold and Pierre Hamburg in 1982. They took the Charlie Strauss model and they added more degrees of freedom in the baroclinic range. And so in this case what happened is that instead of just having the stationary solution some of them were actually fixed points in Chinese and Strauss paper. Basically the baroclinic waves interacted with this large scale flow and they created regimes in which the state of the system was actually orbiting around the stationary states for a while. And then moving to another state. And so these for example are some examples of the various regimes that were found. And one of them for example this one they referred to as sorry. They called it hardly equilibrium. You know the flow is zonal. And then they talked about two regimes that were in fact quite similar to the ones found by Charlie and the war. They referred to as the truck equilibrium or the ridge equilibrium because these were states with well developed planetary waves. In one case the ridge was located over the mountains and in one case the truck was located over the mountains. So this actually was an hemispheric model. It covered so the idea was to explain the whole basically hemispheric circulation. A similar paper was published in 1988 by Vota and Legra and they were actually concerned with just explaining what happened downstream of a maximum of the jet. The goal was to explain the variability and flow regimes in a particular sector. In this case they had mainly the Atlantic in mind. So what they did again they had a channel with a level model and they just imposed a thermal forcing that would create basically an intensification of the jet. Basically on this left side of the channel and then they look what happens downstream of the maximum of the jet. The difference with the respect to the Reinhold-Empire Hamburg paper is that the large scale flow in this case has no multiple equilibrium. But still because of the interactions between the various scales in the model and in particularly the feedback of the high frequency transients basically they could have one regime where the flow was almost zonal. One with an intermediate wave amplitude or a slightly perturbed zonal flow in fact they called them zonal too. And then one third equilibrium where the flow created a strong dipole with actually an inversion of the zonal flow and they called this a blocking. So somehow these two papers describe characteristics of the different approaches that were taken in one case. People were looking at the large scale planetary circulation and how basically they responded to forcing related to geography like the mountains or the Lansing contrast. In the other case somehow the effect of this forcing was just thought to produce a mean state that would be the only stable solution but because of the interaction with the eddies then you could still have multiple regimes. Now in terms of observational evidence of course these are all models. A good question is what happens if you look in the observation. And I mentioned already that Suterra started looking for observational evidence and if you the simple idea is that if you have a model where part of the attractors are more populated than others. Then if you look at the probability density function of the states of this model. This probability density function should differ from a simple multinomial distribution but possibly have multiple peaks, multiple modes corresponding to the different regimes. So in the models where basically the main difference between the different regimes was the amplitude of the planetary waves. Then this was suggest to look for evidence of multimodality in an index of the planetary waves. So this was what Suterra did originally and he expanded his approach in a paper with Hansen in 1986 and he used the at the time NMC analysis. To look at bimodality, he was looking for bimodality in the probability density function of a combined amplitude of planetary waves. And because the theory tells you that these regimes occur because of basically variability in waves which are near quasi stationary. They only use the zonal wave numbers, which theory tells you are quasi stationary in in the non-handsphere winter circulation and they restricted so their analysis to zonal wave numbers two to four. They chose one particular band of latitude so they average the wave amplitude and they computed this probability density functions and you can see them on the left hand side. These are basically examples with slightly different time filters and you can see that these functions are not unimodal. They have two bands, especially this one. Now there's been a lot of debate about the statistical significance of this bimodality, whether the smoothing was enough, whether the sample was large enough. Some people have confirmed Hansen and Souter result. Some people have disputed Hansen and Souter result. So literature about this. So I don't want necessarily to go very much in this debate, but recently David actually tried to sort of compute a planetary wave index. It was a bit more general and the results are actually in a paper that has come out in a recent book published by Cambridge University Press. And so this is the specific recipe. You can follow this recipe if you have a set of reanalysis and compute yourself this index. So you take, for example, daily fields of your potential hype for the winter period for a large set of winters. As we said, we are now looking at large scale regimes on a time scale longer than a planetary wave. So you want to first filter out the high frequency variability. Then you do a space filter. So you only retain the large scale waves. And in some case, David tried to be more general and less restrictive. So include wave number one to five. Then you do your average over a certain band. And David chose 55 to 265. That, in fact, is the latter where you have the largest amplitude. Then you combine, you compute the root mean square amplitude of this profile. And this root mean square amplitude is a single number that will basically is your index for the planetary waves. So once you have this data set, you have to remove the seasonal cycle because of course the wave amplitude has a seasonal cycle. You are left with an anomaly. And then you can compute a histogram showing how many times the planetary wave index lies within a given range of values. And you can also estimate this move for the density function based on this data. Now, if you do that, that's what David found. And again, you may question whether this deep is significant or not. So whether you should say that this distribution is bimodal or unimodal depending on your smoothing. So if you try to fit a very smooth solution, you will get the dotted line. The histogram are actually these colored bars and a sort of intermediate smoothing gives you the solid line. So this somehow shows a problem which has always been debated when we try to find the regime. So how much I should aggregate the data? How much should I smooth them? And how much should I filter? There's a whole literature on that. And again, we cannot just go to all of this. But this is basically one way of trying to condense everything in one single number. Of course, you could say, well, why one single number? Can we do maybe just two? And so this, you can certainly do that. And in this case, one approach is to use, as Angel mentioned, the first to do any analysis of the flow you want to analyze. And then you just look at the leading principle components. And this is a study with Susanna Corti and Tim Palmer, published in 1999. We look again at the planetary scale flow. We look at the first two principal components. And you can actually use non-parametric estimates to compute the two-dimensional probability density function. And if you do that, again, with some intermediate smoothing, you basically find the structure with three maxima. And these three maxima correspond to these three particular anomalies. These two anomalies have opposite sign in the Pacific and North American sector, but both have a positive projection on the North Atlantic oscillation. The third regime is a negative North Atlantic oscillation. Basically, in this direction, you change the North Atlantic oscillation. In this direction, you change the Pacific structure. But you can do that sectorially. And David, again, in the same article, has done that for the first two pieces of just North Atlantic variability. And again, he did that with different smoothing. So actually, he did using the first three. So in this case, you can actually see plots of two-dimensional PDFs using different combinations. So on the left, you see a PDF where x is the PC1 and y is PC2. Second plot, you see PC1 on the x-axis and PC3 on the y. And PC2 and PC3 are on the right. And again, different kind of smoothing. And again, you can argue whether you should, for example, looking at this particular plot, say, oh, well, there are three maxima, or if you use more smoothing, you may say there's just one. But clearly, what this show is that the structure is complex. So the fact that these pieces are linearly uncorrelated doesn't mean that they are statistically independent. So in fact, if you just take basically a cross-section along one particular value of one PC, the resulting PDF will be different. So this actually shows that the attractor is more complex. And although there are issues of how to compute and evaluate these modes, I think most people agree that the structure of the attractor is non-Gaussian. And so you can still look for areas of the attractor which are more populated. So one can generalize even further and say why only use two, but then if you use more than two, it's difficult to compute a probability density function or even more difficult to visualize it. So then the approach is to use a cluster analysis and the K-means method that Angelus discussed in his lectures is one of the most used approach. There are other techniques that have been emerged. And if you actually look at the... Probably there are still on the web the presentation from a school on flow regimes that we had, I think, three years ago here at ICDP. And there were many new approaches to the basically cluster analysis, hidden Markov chain and self-organizing maps and a lot of other modern statistical methods to perform the same. The K-means is very simple, it's easy to understand, easy to code and it's very good that you can actually play with it on the web. Now when this method has been applied either to hemispheric fields or to regional fields, usually people tend to find three to four regimes. For example, Kim Otto and Gill in 1993 found four hemispheric regimes characterized by different phases of the Pacific North American and the North Atlantic oscillation pattern, so different combinations will give you these four regimes. If you look on the Atlantic, these regimes have already been shown by a number of previous speakers. These are the maps from the Kassou paper, but actually the paper which is usually referred to is the one by Michelangelo Italo in 1995. One has to say that these were previously already found by Robert Vota in earlier papers. And in this case, if you look at the top, the two regimes correspond to opposite phases of the North Atlantic oscillation, which is the leading EOF and so there's a counterpart in the hemispheric regimes, but then you'll find more regional regimes. One characterizes basically blocking situation here over the North Sea and Scandinavia. It's usually called Scandinavian blocking, and one corresponds to a very strong ridge in the middle of the Atlantic and is usually called the Atlantic ridge. So basically you can use different degrees of freedom. If you just use one, well, it's very simple. You can do histograms, you can do PDFs, but then of course you have the disadvantage that you condense everything in just one number. You can use more numbers. You can use a wider phase space to look at these regimes. So let's not see how it goes. Now the point is, okay, if you want to use these regimes to understand the, basically the impact or an anomalous forcing like the one that can come from ENSO, come from NJO, as has been mentioned before, one has to think about in general, what is the impact on external forcing in nonlinear system? And you can maybe experiment with some very simple nonlinear system like the Lorentz convection model in 1963. What this was described in the paper by Palmer that Angel mentioned before and in other papers that we did with Laura. And what actually comes out is that if you have a model with strong nonlinearities and flow regime and you basically apply a small anomaly in the forcing pattern, usually what happens is that the regime structure is not changed very much, but what you change is the frequency of occurrence of these regimes. For example, if you do it in the simple Lorentz model, which has a symmetric attractor with two regimes which are equally populated, you put a perturbation in the x or y axis and this will make one of these two wings of the attractor more densely populated than the other, but the structure of the attractor would basically remain the same. On the other hand, it's also a nonlinear system. The number of multiple stationary solutions may depend on a forcing parameter, so if you actually change the forcing parameter substantially, the system can go to what are called bifurcation points, so basically points where the number of stationary solutions changes. And so even in the atmosphere, if you have some strong forcing anomaly, you can actually even modify the number or the patterns of these flow regimes. So can this concept be used for a long-range prediction, both inter-seasonal and seasonal? If you have enough data, you need to have large ensemble because these computations require a lot of data to get statistically significant results. But in the mid-90s, when I was still here, there was a very close cooperation with David at the time and Susanna Corti, and we looked to a very large set of ensemble simulations made with the model of the center for ocean and atmosphere studies. And we looked at the Pacific sector, we did a K-mean analysis, so we found four clusters corresponding to different, basically intensity and position of the planetary waves in the Pacific and North American sector, and we computed the frequency for each winter in the record, and then basically we tried to actually see whether the observed frequency was somehow reproduced by this ensemble. These ensembles were run with observed sea surface temperature, so basically this was an experiment trying to see whether the perturbation induced by different tropical sea surface temperature, for example, in Nino cases, could actually change the frequency of regimes and whether the model could reproduce that. And what we can actually see here, so the blue line is the observation, the red line is the ensemble mean, and the green line is the spread with the ensemble, and you can see that definitely for some of the regime, especially with this Pacific draft, that draft corresponds to the positive phase of the P&A pattern, you can clearly see that the inter-annual variability of the frequency is strongly modulated by the forcing and it was well reproduced by the college general circulation model. Some variability also here in the Arctic law for this period, some of the regimes, like for example here the Arctic high little sign of predictability. And that is the consequence of the fact that the interconnections from ENSO would project preferentially of some regimes rather than others. Now this particular set was in fact large enough, there were 55 members if I remember correctly for each year, and so what we actually did, we were actually able to do a cluster analysis for each single year. And in this case means basically there's a simple parameter that tells you how strong the clustering is or how well aggregated the points are, and this is basically the ratio between the variance explained just by the cluster means and the internal variance of the cluster. So the larger is this number, so the more variance you just explain with the cluster mean, usually the higher is the significance of the clustering. And what we did was actually to compute that parameter for the classification done for each individual year, we just choose a partition in just three clusters. And in the left diagram, you can see this number plotted against the Nino 3 SSD anomaly in this one. And what you see is that there's clearly a negative correlation. So basically this tells you that in the Pacific the strength of the clustering, so actually the significance of these different regimes is actually stronger when you are in the La Nina situation. And progressively when you have a Nino case, the clusters tend to sort of merge with each other to the point that when you look at very strong Ninos, in this case this were 82, 83 and 96, 97. In fact, the parameter become as small enough that the clusters are no longer significant. So this can be interpreted as an evidence for bifurcation. You know, these extreme ENSO events create such a strong forcing for the planetary waves that somehow pushed all the system into one single cluster. There was no evidence within those seasons of a multiple existence of multiple regimes. So in this case this was done similarly, but in case basically plotting the data as a function of the first principal component of the North Pacific variability on the seasonal scale which is actually the main response to ENSO. So again this gives the same message. And if I go to the strong La Nina process, maybe you get a strong forcing that is similar to the Nino, but it is not like this. There's a physical isolation. So when you have a bigger temperature gradient and you shift to the jet, you can see the response at the moment when you shift to the jet for the east wind to the south wind. It pulls the storms for the east wind to the south wind. And what happens is the blocking over the last 15 years tends to some extent to an interaction with the storms, basically. So there's a lot, which is also known at the top of the hill in the area of Nino and plus or less in the North. And it's not a physical connection. And now I would like to come back to what has been only one of the main issues in this school. So the relationship between, you know, subsaison of aerobics and specifically the MGO, the phase of the MGO and the occurrence of regimes in the North Atlantic. You have already seen this picture many times. So you know what it is. And so what I'm trying to propose is one possible interpretation of this dynamical link based on one category of models for flow regimes. And in particular, I will use the papers as a reference for a dynamical framework. The papers on thermal equilibration of planetary waves basically describe the variability in the phase of planetary waves with respect to the surface temperature distribution. So I already mentioned the Michelin, warm paper, papers by Glenn Schatz and Marshall and so later on also approached this problem. And somehow if one looks at the observational counterpart for this theory, probably, you know, what comes to the mind is the so-called cold ocean warm land pattern. The pattern that was described empirically by Mike Wallace and collaborators in the mid-90s, which is basically a pattern which describes the variability of the large-scale atmospheric flow with respect to the surface temperature. So in the sort of cold ocean phase, so you have cold air over the warm ocean during winter, the opposite phase will actually have warm air over the ocean. So in a paper in 2001 that we did here with Fred, David and another colleague of ours, we tried to relate basically the properties of the cold pattern to the predictions of thermal equilibration using the speedy model. And we tried to find a simple index that could somehow quantify these interactions. And what we tried to do was actually, since one of the predictions of thermal equilibration theory is that the switching of the phase will actually change the thermal forcing of the planetary waves. Because in one case, if you put cold air over the oceans, which are relatively warm in winter, the difference in temperature would be very strong. You will have a lot of heat transfer from the ocean to the atmosphere. If you do the opposite, if you put the warm air over the ocean, then the difference is reduced and the fluxes are reduced. So we decided to use the surface heat fluxes as a measure of this transfer and to see how other variables related to that. So we defined an index which is basically an index which is positive when you have a positive transfer of heat from the northern ocean to the atmosphere. So you have upward heat flux over the oceans and downward heat flux over the continent. The opposite phase is related to the so-called equilibrated phase where actually you have warm atmosphere economically located over the ocean so the fluxes are strongly reduced. If you compute this particular index, for example, from the era interim analysis for each winter, 82 to 2001, you get quite a lot of interannual variability but also some decadal variability. You can see, for example, in the last ten years or so the values are predominantly negative. The covariance of this index with geopotential height gives a very strong wave number two pattern. So we are expecting this. So in the positive phase, you actually have low geopotential height over the ocean and so cold lower tropospheric temperatures. And you see here basically a strong North Atlantic oscillation signal. This is actually the map associated with the surface heat flux. In this case, it's defined positive downward. So basically this tells you that in the positive phase the oceans are cooling because they are transferring heat to the atmosphere in the northern latitudes and there's some compensating warming in the subtropics. So this is what you get from the observation and so basically you can say that this is a pattern which is just associated with different planetary wave configurations related to a forcing which is actually originated in the extra tropics. Now you can take a different approach and go to the tropics and say, well, what about the forcing associated with these different heat sources along the equator? And this is a pattern that again was actually diagnosed from the seasonal prediction, but actually if you look at the structure of the pattern in the western Indian Ocean and over the maritime continent this looks very much like some of the patterns you see in, for example, phase two of the modern and Julian oscillation. So if we actually just focus on this region, which is actually a region which is important both for seasonal and intracesional variability and we look at the teleconnection with geopotential height in the extra tropics so you compute an index and you look at the covariance map with this index once you standardize it, then you get a pattern like this and again you see a low over the North Pacific and a positive North Atlantic oscillation signal. So this is somehow consistent with some of these MJO teleconnections so it tells you that if you have increased rainfall over the Indian Ocean, drawing over the maritime continents and the West Pacific this would be associated with the positive phase of the North Atlantic oscillation. Now if I actually show you the pattern that was in the previous slide so the one that was actually associated with thermal forcing originated in the mid-latitudes and if you compare these two maps they clearly have quite a strong degree of similarity and so somehow one way, one possible way of actually interpreting this particular teleconnection is to say that some rosby waves that will propagate from the tropics into the mid-latitudes will actually stabilize one of these two opposite regimes of the planetary waves. So you have some sort of forcing which is the thermal contrast in the extra tropics. This may give different kind of equilibria and what you do when you change the forcing in the tropics you send some signal in the extra tropics that will push the system into one regime or the other regime. So this is basically one example, one can find many others of how you can actually use the sort of simple linear teleconnection concepts that has been used many times to study the impact of tropical forcing on the extra tropics. You can actually relate it to the variability that is actually originated by nonlinear dynamics in the extra tropics. So we can come to the conclusion so what we have seen is that circulation regimes represent an important dynamical feature of many nonlinear models of the atmospheric circulation. For regimes in the extra tropical circulation a balance occurs between the dynamical tendencies of the large-scale flow and the nonlinear feedback from high-frequency bioclinic waves and that comes if you average on a time scale longer than about 10 days. So this is a bit different from what Angel has shown because of course you can use the cluster analysis to look to more sort of fine grained structure that as David mentioned was so much related to this large-scale equilibration but on the other hand have the advantage that if you want to use them as predictor of local events then they will be more strongly correlated with local phenomena. So you can use both and Laura will also show in her lecture some examples of using this concept on a larger scale to do prediction. So regimes have been detected in long records of either analysis or model data using one or two-dimensional probability density function or cluster analysis applied to either atmospheric or sectorial domain. The anomalous forcing, for example, from ENSO the MJO can affect the property of regimes by modifying their frequency of occurrence if the forcing anomaly is not too strong or it can even change the number of regimes if the system goes across a bifurcation point and if you have a large enough set of model data you can actually detect both kind of behavior. So as one example of these interactions between signal propagating from the tropics to the extra tropics and this interaction with the regimes I've shown you this example of basically looking at the pattern that basically is associated with different phases of the planetary waves as a response to thermal forcing and you can actually see that the forcing originated from the Indian Ocean and the maritime continent actually has quite a strong projection on these particular patterns. As I said, this is one example and you can find many others but I think that for this morning please stop here.