 To je pomembno, da ga se v tem, kako se vse nekaj ne vse, na papani in benzilu, tako v taj nekaj gaden. Tko je tudi osrednje tem, nekaj nekaj, ki se počutite na predstavku. Srečo, ga se počutiti o atlantici režimi, Moj imam vse, da te vse zelo, da se vse vse zelo, na vsevručnih modeli, in da se zelo vse vse vse tega zelo začelti in zelo in začelti in s trapečnimi. To je to, da se izgleda, in da sem izgleda, da se lahko trafimo, da haveem naši dynamik, naši dynamik, za to, da je zapredal, these regimes. We found this cluster, but is there a dynamical reason why this cluster should be separate. Now, in the past people have looked at quasi-geostrophic models and they found things, basically similar to what Wojtar and Legra found. možemo skupati vz newspapersu v Atlantiju, especially v 진 solo na kratahvě po vlasti, zaznjah na tom, da je vse povedet v español punchingesiteginje, ali pomožneje ta ljud, ampak od nitinga, da je to vse mene. Vzumila, da je, da je, nekaj in atlantij, čim vječenem, ki so poveden, smosta, katerovi cilje, kana da, in tudi vzvečen, vzvečen, v zelo v Westem Bandriči. Zato je to, kako to je. Zato je, da je teorič na termolekuliviracijne, plenitari vzvečen, ki smo vzvečali, in ki so, da je vzvečen, kako se vzvečen, kaj je. Zato smo vzvečen, zelo, da je zelo vzvečen, da je, da se vzvečen, zelo, da je zelo vzvečen, vznikno, in vznikno. Zato tudi, da smo pričočili, je to, da je veliko sempljeno model za izgledanje vznikov izgleda. Zato, da smo pričočili, da izgleda je izgleda, je to poddajte izgledanje. Zato, predstavno, je, da se ne moguš vznikov, ali, da se vznikov vznikov izgleda, in kako se v realnih veči. Prvno je tudi, da vse predstavimo nekaj spesivnih procesov, kaj je vse režim veči. In, kako smo videli, da vse režim veči, da imaš režim veči, da imaš balans vsega pozitivna vsega in negativna vsega, ker je vsega vsega in stabilitiva. So in order to have a bimodal distribution, it means that when the system is around here, there will be some instability that will bring the system either in this direction or in this direction. And on the other hand once you are in this point, if you move further down here then you need some restoring force that will bring you the system back. So you need an instability and some damping, some dissipation terms. Now in linear systems, if both the unstable and the stable part are linear, they will combine and the result will be that either you have an unstable mode or you have a stable system. In nonlinear system you can make this feedback dependent on the state itself. So in particular if this instability actually decreases gradually, then basically you reach a state where the restoring force may win and bring the system back. So you need these two ingredients. So the first thing was somehow to try and find which are these two ingredients that matter for the North Atlantic oscillation. Ok, so I employed the same sort of covariance techniques that I had used for the teleconnection studies that I showed before and I started by defining an NaO index. And since I think my interest was to basically look at NaO in the context of this possibly this wave number two pattern, what we did was not to use maybe the traditional Iceland minus Lisbon index or local ones, but something that would really look at the structure of the planetary waves on a wider scale. So we defined the index as the difference in geopotential height between two boxes, basically one covering this area here and one covering this area here. If you do that, you have an index of the NaO, you standardize this index and you can compute covariances with various fields. So let's start with the simple ones, geopotential height at 500. Well, this is just the result of how you have defined the index and you have a pattern that looks like the traditional NaO definition. Now, if you do that with geopotential height at 850 instead of 500, well, you get a pattern which is very similar, almost in phase, slightly smaller amplitude. And if you look at the difference between the two, basically that gives you the mean temperature in the lower troposphere and that is what you get. And again, perhaps you see a bit more structure here, but roughly the position of the low and high here is roughly similar. So this tells you that this NaO anomaly has roughly an equivalent of a propic structure. So the temperature and the stream function line are in phase with each other. Now, you can also see that most of the signal is actually in this half of the hemisphere. So you can draw a line at 135 west and 45 east and actually look at what happens in this part of the world. And a justification for this is that if you actually do a zonal mean of these fields, then what you get is the blue line. So for example, you see that the zonal mean geopotential gradient is increased during a positive NaO between roughly 35 north and 65 north. And so you have an increase in the strength of the zonal wind in this region. And you have opposite signals to the north and to the south, similar thing at 850. And you can see this mainly comes from the temperature. In the temperature you see basically an increase in the... So you have a stronger zonal mean gradient of temperature so temperature decreases faster with latitude during a positive NaO. Now, what are the green lines? The green lines are what happens if you only consider the contribution from this half. So if you just set the other half to zero and you can see that you get almost the same thing. So this amount justifies the fact that we can study our north Atlantic oscillation basically just concentrating on what happens in this part of the world. So if we think about something which has roughly zonal wave number two structure when you look at the whole hemisphere, then it becomes basically just a wave number one in this half of the hemisphere. Now, an important thing about... I think this is the real crucial point of the whole theory. An equivalent barotropic anomaly cannot affect temperature because the stream function lines and the isotherms coincide. So you have, if you have a low, you also have a cold anomaly. So the flow just goes, follows the isolines of temperature and by definition you have no temperature advection. So in order to have transport of heat by the horizontal flow, you need to have a baroclinic structure. So one where the stream function and the temperature are not in phase. Now, what happens is that... Sorry, we are here. So the NaO anomaly is roughly equivalent barotropic, but this NaO anomaly is superimposed to climatological stationary waves, which are not equivalent barotropic. So if you actually plot... I took the stationary waves at 150 ektopascal in DJF for this later 30 years or so, and what you see here are a plot of meridional velocity, V, and temperature. Now, if the structure was equivalent barotropic, then this would be out of phase with each other. But actually you can see that there are regions here, here, here, here where they have the same sign. This means that if you do the product of this field, these are only the eddy fields, so the stationary waves. So if you multiply this by this, the result is positive. So this tells you that the stationary waves transport heat to the north. So when you think about the effect of an NaO anomaly, even if the NaO anomaly is equivalent barotropic, it can still change the heat transport because it's superimposed to stationary waves, the rest of the stationary waves. You can see here that while the NaO anomaly, you remember, has a dipole in this direction. These stationary waves have a broader meridional scale, so you have the same sign almost from 17.0 to 20. So what happens if an equivalent barotropic anomaly with the dipole structure in the y-direction interacts with a baroclinic wave, which is baroclinic, but it has a broader meridional scale. So here is our idealized version, so you have, now we are just looking at this one, 35 West to 45 East sector. So this, at the top, you have an idealized representation of temperature and V in an equivalent barotropic anomaly with the dipole structure, like the NaO. So this is our sort of NaO-like anomaly. And then you have the stationary waves with the broader meridional scale, like this one. And while the equivalent barotropic anomaly has the V and the T out of phase, we have seen that in the real climatology, the V and T actually are positively correlated. So I have only put just a 15-degree shift between these fields. So these are idealized pure sinusoidal functions. Now, what happens if you compute the product? So this is the result, so basically you can decompose the anomaly in product of the anomaly in V times the climatology in T plus the climatology in V times the temperature anomaly. And because of this particular structure, so while, again, in an equivalent barotropic anomaly these two terms exactly cancel each other, because the climatology is not equivalent barotropic, the VAT-clim term dominates. And so the result is that in the total result is that a positive aneo anomaly would actually increase the heat transport around 60, no, between 60 and 70 degrees north. So if we do the zonal mean of this, you get this curve and this curve has two zero points. Why it has two zero points? One is that because the climatological temperature at the goes to zero here, so that gives you this zero. And this zero curves because the V goes to zero in the middle of the channel, the V of the anomaly. While around 65 to 70 degrees north you have these strong peaks that dominate the negative ones and you have this. So this is the V star to star. Now to get the temperature tendency you need to take the wide derivative of this. And so if you look at this part here is zero, here is zero, so the mean derivative in the southern part of the channel is zero. While the derivative in the northern part of the channel is positive and very strong. So this means that the aneo anomaly will hardly change the temperature in the southern part of the channel but it will cool the northern part. So if you cool the northern part and you don't change the southern part you increase the zonal mean wind. So basically what we are telling is that the eddy part of the aneo anomaly is able to reinforce the zonal mean part of the aneo anomaly. So an eddy will actually the eddy configuration that observation is found together with the increase in zonal mean wind is actually able to increase the zonal mean wind by transporting heat to the north. Now this is an idealized case. Now what happens if you look at the real world? So I've just left the V-star, the star panels there. So these are the equivalent maps derived from observed data. So again you do the covariance of the aneo index with V, you do it with T, you do the product and this is what you get. And again, so this is the product V-star from the aneo anomaly times T-star from the stationary waves. This is the other term, which is smaller. The result is dominated by this term and the curve of the meridional heat transport has these two zero points and the maximum around 65 to 70 degrees north. So the simple framework explains what is the effect of the aneo on the zonal mean flow. So this gives us our positive feedback because you know you have an aneo anomaly in the eddies, this actually is able to increase the zonal wind which is also associated with the positive aneo. Now we need negative feedback. Negative feedback is provided by the heating associated with the surface in fluxes. Now we can actually write the turbulent heat fluxes which are the some more sensible and related heat flux basically as a flux of more static energy at the surface. And the standard aerodynamic bulk formula will tell you you have a coefficient you multiply it by the density of the air and you multiply it by the wind speed at the surface and then the difference in basically more static energy between the surface and the atmosphere just above the surface. Now usually in many simplified models you can actually represent this heating by neglecting the variability in surface mean wind. You say well that's climatological, you are fixed. And so you can probably neglect the the thermal associated with latent heat and then you have a simple linear damping of temperature. This looks fine, but actually if you look at what actually happens in the North Atlantic you find that the surface is fluxes are mainly driven by variation in the surface wind speed. So if I now compute the covariance of the 10 meter u wind with the NaO anomaly then you get well what we expect. So stronger zonal winds basically in the mid-latitudes. So west of the anomaly here east of the anomaly in the tropics. The corresponding variation in the surface heat fluxes is this. And you can explain a lot of this structure by just basically adding a positive anomaly in climatological surface winds. So basically wherever you have a positive anomaly superimposed to positive westerlies you increase zonal wind and you increase the heat fluxes. In this case the heat fluxes are upwards. When the positive anomaly is superimposed to climatological Easterlies you have a negative. When the anomaly Easterlies are superimposed to Easterlies, climatological Easterlies the trade winds again you have positive. So this is a famous triple in heat fluxes that drives the triple in sst anomaly which is associated with the NaO. Now this is the covariance with this NaO index but you can basically get almost the same pattern if instead of doing the NaO index you just take the mean zonal mean 0.850 as your index and you basically correlate this with the heat fluxes and what you get is this. So basically it tells you that over the Atlantic if you know the strength of the zonal mean westerlies you basically know the intensity and distribution of the surface heat fluxes to a pretty large extent and that's because we did in the heat fluxes. So the heat fluxes will put heat into the atmosphere and this is again the temperature anomaly at 850 which is associated with a positive NaO and if you actually compare the temperature with the heating you see that these two anomaly are negatively correlated. You can see here there are some profiles so the red line is the heating produced by these heat fluxes the blue line is the temperature anomaly and okay there's not an exact anticorelation but clearly there is an anticorelation which means that this gives you a negative feedback so you increase the zonal wind you increase the surface heat fluxes but these heat fluxes actually project negatively onto the NaO anomaly and this will give you a negative feedback. Okay, so now we have the ingredients we have the positive feedback and the negative one and we can try to put them together and so again we'll try to do it in a simplest possible way we have a channel that covers the sector that I've described and as in many lower order models you just have three degrees of freedom one for the strength of the zonal wind and one for the wave Now the difference with respect for example to the Charne de Vore is that the Charne de Vore channel was actually a closed channel so the wave was designing such a way that you had zero meridional fluxes at the boundaries which may be desirable but actually here the crucial point is actually what happens at the boundary and the NaO anomaly is the largest amplitude at this extreme so this is what I would call an open channel and of course so you might argue in open channel things can also come from other things may come from the boundary so that's why we call it a heuristic model because we assume that it's fine to describe what happens in the channel so if you believe me then basically you want to describe what happens to the stream function so you have a mean zonal flow minus ui you have a wave pattern which is associated with the NaO which is described by this wave and in particular this phase this particular phase of the wave has this low here over Greenland so it looks like the NaO anomaly and this actually is more in phase with the forcing with the thermal forcing because at high latitudes it's cold over Canada and warm over the ocean so the climatological lency contrast will project on to this pattern but what we want to know is what happens to this phase which looks like the NaO then what we have to consider are the interactions of this wave with a wave of larger meridional extent so the wider scale component of the stationary waves this is important because this will give you the feedback on to the mean zonal flow so the first thing is let's just consider the first element that is just the advection linear advection this is standard planetary wave theory you have just one mode so the mode does not have interactions with itself so it just interacts with the zonal wind so you can use you can basically describe it in evolution by a simple barotropic vorticity advection where the laplacion of the stream function is the vorticity and its derivative is basically given by the advection of relative vorticity by the zonal wind plus the transport of planetary vorticity which is the better term simple standard you find in all textbooks hold on whatever so because your wave is a simple wave it's an eigen function of the laplacion you can further simplify this equation and so the evolution of the stream function is basically proportional proportional to its zonal gradient multiplied by the difference between the two winds one is the actual zonal wind and this is the value of the zonal wind that will make the wave stationary so the wave will become stationary if this is zero so this beta over n square where n square is basically the total wave number of the wave when this is zero the wave is stationary so if we now redefine if we define a u prime variable as the difference of u from this stationary value and you also assume a time scale of about 23 days that will show you why basically if you project this equation onto the simple mode you get some very simple cycle ok so the a is increased by the reduction of b and b is decreased by the reduction of a so this will give you a simple propagation and this factor alpha depends on the time scale you have assumed so we can choose this time scale in order to make this alpha coefficient a and b are the two phases of this wave so a is the coefficient that multiplies the nao pattern and b is the coefficient that multiplies the pattern which is just in phase with the lensy compass they are exactly and they are at 90 degrees so ok so we start from here but then we have to add more terms and we are seeing that what we have to take into account is the divergence of the meridional heat transport the thermal dissipation due to the heat fluxes and in addition you will have some radiative damping ok so we have seen that the divergence of the meridional heat transport is a positive feedback so if the nao is in the positive phase so if a is positive then the u wind is increased so in the equation for u prime you get a multiplied by a positive coefficient then we have to reproduce the effect of the fluxes the fluxes will be somehow proportional to the zonal wind anomaly but they will act as a damping so this h term has a negative sign in the equation and then you have some simple relaxation towards an equilibrium, radiative equilibrium now at this point you need to estimate this gamma and sigma coefficients and you can actually do it from the covariances that I have shown you one can actually compute assuming the scaling in the u what values and also for the radiative damping you can basically estimate the anomaly in long wave radiation associated with the given temperature anomaly and if you do that I have no time to go to the calculations you end up that gamma is roughly equal to 2 sigma is also roughly equals to 2 while the radiative damping is about 0.5 which means that basically the surface fluxes give you an anomaly of 40 watts per square meter the radiative anomaly will give you a flux of about 10 watts per square meter so the radiative time scale will be four times so the time scale is basically the inverse of these coefficients so this basically since the time scale here is 20 days in order to get rid of this alpha term if you have time scales for the heat flux conversion and the heating by the surface flux is of about 10 days this will give you a coefficient of about 2 A, yes the u is the meridional gradient the b is also a stream function difference in that case you don't need A and B just you just relate B with a 90 degree phase yeah, but the revolution is still so basically you have a cycle so if you go back to the in fact this is what this equation tells that this is just a simple cycle between two phase of the waves that are at 90 degrees so what the wind does is that it just moves this wave downstream so basically this is the result of what you are saying that because the waves are at 90 degrees from each other then you have this simple oscillator which is driven by the direction ok, so now we have put these other terms in and the interesting thing is that if you do some simple transformation so first of all we can say ok we can actually check that's another important element here so in the equation for you there is a u star so we assume that there will be some relaxation towards some climatological state however we have written the equation in terms of the deviation from the stationary value so if the climatological state is almost identical to the value that will make this wave stationary then it means that this u star is zero now if you compute what the stationary value is for this wave number 2 it comes out to 12.8 meters per second if you actually look at what the zonal mean wind in this channel is at 500m bar height from error-intimary analysis you get 13.1 so they are really very close so you can basically assume that this justifies our model because if this particular wave is close to stationarity it means that it is quasi resonant and it can display a large low frequency variability so we can set u star to zero gamma is roughly equal to sigma we have seen from observations and so if we define b star as the difference between b sorry b prime difference between b and b star then you get this particular system of 3 equations and if you remember what I showed you for the Lorentz model it's basically formally the same system because basically what I assume is that because I've chosen I've chosen b I've chosen the phase of this wave so that the climatological state is just in phase with b and not with a so basically b is something which is in phase with the lensi contrast and the NaO is actually orthogonal especially orthogonal to the lensi contrast so the lensi contrast will make a strong forcing on b but will not make by construction a forcing on a so this comes out if I had chosen a different phase then you would have had forcing on both a and b but simply because I've chosen the phase in such a way that just b is aligned with the lensi contrast then you don't have an equivalent a star now if you make this transformation you end up with the system which is that looks formally equivalent to the Lorentz model and this b star minus sigma plays the same role as the R parameter in the Lorentz model so basically the is the parameter that forces that determines the amplitude and the separation of the stationary solution so basically it tells you that it's the forcing in the in the planetary way field which projects on b that is the factor that determines the difference between the stationary states the only difference is that this this radiative terms come they are different between the different variables in the Lorentz model while we have basically the same coefficient in all three equations but ok, so if this is formally equivalent then you can expect that the attractor is similar in a suitable range of parameters and well if this b star determines the amplitude you have to say what is b star now it's very difficult to say what is b star because it's a cumulative effect of thermal forcing but also holographic forcing of the climatological waves so here we took an empirical approach we can compute what the stationary solutions are as a function of b and we know what the amplitude of the typical nao anomaly is so we choose a value of b which is consistent with that it turns out to be 12 so if you choose 12 and you integrate the system well you would expect that you get a Lorentz attractor so basically and the green points here so these are basically scattered diagrams of the u-wind against the say nao amplitude this is the u-wind versus the wave which is in phase with the lensi contrast and these are the two waves and what you see on the left are time series for the three coefficients ok, it looks like the Lorentz model so it's fine so this will basically tell you that you will have two nao regimes one corresponding to stronger zonal wind and the positive phase of the a wave and one corresponding to weaker zonal wind and the negative phase of the wave and the green points are given by the stationary solutions well there's one stationary solution which is again zero motion like in Lorentz model and then two stationary solutions which are weakly unstable ok the point is what happens if you change b that's interesting but that would bring us to a lot of other lectures now what this model describes basically just the interaction between the planetary waves and the zonal flow now people may argue well most studies that point out that in the Atlantic the regimes interact strongly with the baroclinic waves and in fact if you actually think at the traditional view of the atmospheric energy cycle the view is that basically the energy transformation are mainly driven by baroclinic eddies there are no stationary waves in this scheme even if as we have seen planetary waves transport heat and convert energy so the traditional view is that you put diabetic heating you create zonal mean available potential energy somehow the forcing of the circulation will actually turn this zonal available potential energy into eddy potential energy then the baroclinic eddies will convert this eddy potential energy into eddy kinetic energy and then when the baroclinic eddy decay then this kinetic energy is transformed back into kinetic energy of the zonal mean flow so there were a couple of very nice papers by Martin Ambaum and Lenkanova they were actually presented here at ICDP about one year ago during the open IFS workshop and in fact I have to give credit to them because it's the heuristic model that gave me that challenged me to find this simple nao model basically what they were interested in was in a much simpler problem so just interaction between the zonal mean flow and the integrated amplitude of baroclinic eddies and what they developed is a simple non-linear oscillation which is analogous to the predator of prey model where basically the amplitude of the eddies grows at the expense of the available potential energy of the zonal mean flow and on the other hand the growth of the eddies due to barotropic instability is proportional to the baroclinic itself so high baroclinic makes the growth of the eddies faster but in the end the eddies extract at the sorry, extract zonal available potential energy so the baroclinic decreases and then you have a nonlinear oscillation and what they've actually estimated the leg between the growth of the of the baroclinic eddies which was quantified by the heat flux the material heat flux and the baroclinic which was basically index of static zonal mean static stability and they found some leg so the eddy growth legs by after a couple of days the maximum in baroclinity and you have a nonlinear oscillation so if this is an important mechanism and as they have argued you might say well your model doesn't take this into account at all so what happens if in fact there are baroclinic eddies that go at the expense of the zonal flow so the minimal way to do that is that we can actually so far we have assumed an equivalent barotropic structure so our zonal mean flow was just determined by one number but you can be a bit more specific and they compose the zonal mean wind into just the thermal part which is proportional to the horizontal temperature gradient and that is basically what we were talking about before and a fully barotropic part which is the same at all levels and so at the surface the zonal wind will actually be just the barotropic part and so in this energy cycle it means that the growth will the eddies, baroclinic eddies will go by extracting energy from the zonal mean available potential energy so they will decrease this component but then put back energy into this component of the flow so you need one more variable here and you need one more variable here now there's definitely not the time to discuss in detail all the coefficients of this model but basically what we have done is that we have added an equation for the barotropic part this basically tells you that the barotropic part grows by the kinetic energy which is put by the decay of baroclinic eddies and then is done by surface friction and the eddies actually grow by extracting energy from the thermal part of the zonal wind and decay by surface friction and the barotropic conversion into the zonal mean flow so these are basically two nonlinear oscillations like the ones that Ambarov and Novak have developed so the question is if you add these processes will the attractor be totally destroyed just be turned into a simple oscillation or not and well I don't have time to go into the estimation of these parameters in fact it was done quite empirically but the result is no so the attractor is likely modified so again these are the scatter diagrams these screen points just for reference are the stationary solution of the previous model so you can see how the attractor is actually shifted with respect to its original position but roughly the variability so you still have the two regimes what you have of course is more variability on sub-seasonal scales and you have a variability in the zonal mean wind first the thermal part grows then the eddy grows and then the eddy decay and the barotropic part grows so you have actually three lines in the top diagram these again are the two phase of the planetary waves that have a bit more wiggles but still they display this regime and so actually we can look at the relationship between these the zonal mean at the eddies in more detail first of all we can plot the amplitude of the eddies as a function of the thermal part of the zonal wind so basically the baroclinity and as we expect well on average there is a positive relationship so when there is more baroclinity you have stronger eddies but if we look at the detail as the time relationship this is the basically the lag correlation amplitude of the baroclinic eddies and in red the thermal part of the wind and in green the barotropic part of the wind and the axis x axis gives you the lag between the wind and the eddies so this basically tells you that the eddies get larger about three days after the peak in the baroclinity and then they start to decay and then they put back energy into the barotropic part so the barotropic part of the wind as its maximum about five days after the peak in the baroclinic eddies and you have an oscillatory behavior because you basically have two nonlinear oscillators like the one by Ambar and Novak ok, so we didn't want somehow this part it's basically just to demonstrate that this regime structure is not actually destroyed if you add interactions by baroclinic eddies of course you could choose another way of specifying this interaction but since this has been not quite discussed in the literature we just wanted at least to have this type of interactions in the model so basically what we have seen is that the regimes in the NAO can exist because of the balance of a positive and negative feedback that involves the zonal the eddy component of the NAO anomaly and the heat fluxes the positive feedback is due to horizontal heat transport by the wave that interacting with climatological stationary waves which are baroclinic can actually modify the transport at the northern edge of the channel and therefore can either increase or decrease the strength of the thermal wind the negative feedback is due to the thermal damping caused by the heating anomalies driven by surface fluxes and because the surface fluxes are so dependent on the zonal wind speed again this becomes just a simple feedback between the strength of the zonal wind and the thermal damping of the NAO anomaly so adding these two feedbacks to a simple cycle we just described the rotational action of the wave then you get the three variable model which is formally equivalent to the Lorentz model and therefore like the Lorentz model it has a chaotic attractor with two regimes and that's nice so we can say well that's some justification for assuming that positive NAO and negative NAO are real regimes and see if in addition to zonal wind and planetary waves you add the effect of the bioclinic eddies so we have used a simple theory developed in previous papers to do that and basically the good news is that it doesn't destroy the chaotic attractor it just puts basically more variability in the subsesional scale so that's almost work in progress so we still have to finally get it into the literature but we thought that well this is a school of multiple equilibria and so this was quite a fitting topic for that again it's a very simplified model you can say well it's a gross simplification of reality as any simplified model is but at least it gives you a conceptual framework that somehow justifies these observational results the other thing that they can do is that because the calculation that are down to estimate for example this gamma and sigma coefficients these have been done on real data you can do them on models so for example instead of just using these simple metrics that we used to see whether our sim in 5 or sim in 6 models give a good representation of the mean in all the variables we could actually compute these coefficients from the models and see whether they have the right strength of the positive and negative feedbacks and because the regimes come out because this gamma and sigma are both similar and they are both too so it's their balance that gives these regimes if one is dominant over the others you will get something very different so in the future we hope that we could actually repeat these calculations of the strength of the feedback maybe from model data it might become boring I don't know whether we will get anything out of it this could be a way of actually hopefully seeing whether the models have the right dynamics to reproduce DNA regimes and that concludes my lecture so thanks a lot for your help thanks