 I'm actually pretty sure that most of you know more about equatorial waves than I do. I mean, you are in the S2S workshop, so this is not really... So if anything occurs to me, errors, please tell me maybe, in fact, any one of you could give a better presentation on that, but it could be good to remind us what these equatorial waves are, and I will try to derive them according to this simple textbook of Houlton in the most easy way possible, reducing the equations as much as possible and hitting very hard the real atmosphere by doing this. So what is what would be the motivation? If we have a field near the equator, let's say it could be a wind field or temperature or I think what many people also use is the outgoing longwave radiation because we have good measure satellite measurements from the 1980s onward, whereas for all the other fields, the direct measurements in the equatorial regions may be not so accurate over large parts. So if you do that and imagine you have a field, you have a field that let's say velocity from x, y and t. Okay, we can look at the variant of this field in spatial space and get some idea what's happening, but what is interesting to do is sometimes to do a spectral transform of this, to transform this into a field of the velocity let's say head as a function of, and in this case it's not actually a function of y, it's a function of the zonal wave number and the frequency or we will call it nu. So we do a spectral transform, a Fourier transform into this space and then analyze the variance of this square essentially. If you do that in the tropical regions and again I guess everybody of you is more familiar with this diagram than I. So if you look at that, then you draw this energy in terms of the frequency nu and the wave number k, the zonal wave number in this case, we are averaging this field or summing the spectra around the equatorial region. Then something striking occurs to us, we can see some increased energy in some bends in this wave number frequency region and there are these kind of straight lines popping out here and there are some other things appearing here and of course you know what these are. What are these? Can we wait and what are these? I knew that you know better than me. So what is this here? It's the MjO. Okay I guess this is something like mixed velocity gravity waves or something like that. So we may ask the question, can we get a theoretical grip on these waves? Can we try to understand them, this power spectrum? Can we try to assess how we can maybe derive this a little bit from equations and then therefore maybe understand a little bit more from this? I just want to show you from my own work this is this model that I'm running, outputs from this model and here I'm doing something in this experiment. I'm prescribing a sea surface temperature anomaly in the equatorial Atlantic in a switch on experiment. So I'm starting from, I compare an experiment with a control experiment and this anomaly is switched on. So at time zero in this diagram is shown the the wind at upper levels trying to take the Pascal average of about the equator and time is running here in days and here's the the longitude and we're here in the Atlantic zero and at time zero there's nothing no perturbation in the fields but then something is happening the SST anomaly, the sea surface temperature anomaly will create a heating in the atmosphere and they will create waves. Somehow the heating at some point creates a vertical motion this vertical motion hits the tropopause and creates equatorial waves all kind of things but also equatorial waves and also here if you follow these lines here these moves to the to the east and these moves to the to the west but with different as you can see the phase speed could be derived by by looking at this envelope here and the phase speeds are very different. So what are these waves? We can also interpret them as some kind of waves we want so these would be again the Kelvin waves that moves to the east and these would be the equatorial Rossby waves that we see here. In a model their waves may not be exactly the ones that were observed but something similar. Okay so so these were the two motivations for deriving this so this will be kind of a this is kind of a really a master level lecture but I guess it's anyway good to remind of as what is going on. So what I'm trying to do here is to derive these these these spectra in these waves using the shallow water model which is the simplest model you can use to derive these waves. The shallow water model has almost nothing to do with the real atmosphere okay because it assumes constant density no divergence many things horizontal velocities that are independent of height many things that are not true in the real atmosphere yet this very simplified model can be used to derive some properties of these waves. There will be a problem coming back in the end because we're using we're using in the shallow water models parameters that we have no idea what scale to what what what number to put with these parameters and these parameters in the end we will have to adjust according to the observed phase velocities because we cannot really derive them from from observations of this quantity because this quantity doesn't exist in the real world in the real atmosphere. So so again these are the the the assumptions of this shallow water model which is a really hitting the atmosphere extremely hard but it's still surprising that the solutions of these equations are even you know we can derive something analytically which is which is the reason why we are trying to use these equations and some of the solutions look like the observed properties at least you know disregard the vertical structure but still the solutions are even more complicated than we would expect the solution of these equations. So we assume that we have an incompressible fluid with a constant density that that the flow that we are considering is is such that the horizontal velocities do not depend on height. This assumption is of course wrong in the tropics in particularly in the exotropic sometimes you make this assumption some we call this barotropic model you find those kind of perturbations. In the tropics this is usually exactly not the case particularly imagine the case of a heating in the tropics and so induced heating we get a surface convergence and at upper level divergence so no way the horizontal velocities could be independent of height. So in the interpretation of what we derive is then we interpret it to be valid not for the whole troposphere but maybe either for the upper troposphere or for the lower troposphere but in order to make a connection between the two one has to to make some additions that that will come back have you ever heard of the Gill model the Gill responds to an atmospheric heating some of you have heard that also there this this problem the model is based on the same equations in this problem will come back very very severely because in the Gill model we make the same assumptions that we do here so this is very very great assumption so essentially we are starting with the with the horizontal equations of motion and with the hydrostatic approximation there's still not nothing yet wrong about these equations because these are these equations are valid okay then you remember if you do the if you write down the continuity equation and you assume that the density is constant this is the divergence free equation of course just to remind you how do you arrive the shallow water equations we integrate the hydrostatic equation vertically from from an arbitrary height to the top of the medium so we assume that we actually have a have a fluid that can change which the height of the fluid can change and variations of this height gradients in this height can then determine us flow within the fluid that's the very simple assumption if we if we use these equations so so what we get is we can transform the pressure gradient force with the hydrostatic equation and something that looks like this the gradient of the of D times the height of our of our shallow water so if we insert this in the equations okay we get we get this so in principle we are now we have we have two equations the two horizontal equations of motion so by the way what are what are the what are the terms here the local change of velocity this is the advection of the of the velocity the pressure gradient term we we named this already and the Coriolis force the Coriolis force of course is interesting because one may naively believe close to the equator it's zero and therefore we can set this to zero but it turns out that the Coriolis force is extremely relevant for the equatorial weight too so what we can do now is to to use the divergence free so yeah the problem is that we have two equations and three variables still because there is this geopotential height or the height of the of the medium that we we need something else to close the system and that that is delivered by the continuity equation which again we can integrate vertically and derive from this an equation for the height of our system so the the height here of our system fulfills the continuity equation so with this last equation our set of equations or our shallow water equations complete and these equations again have nothing to do with the real atmosphere but can be used to derive some properties of the real atmosphere what we of course we would like to to solve this equation the set of equation analytically in order to understand something we could integrate them numerically starting with some initial conditions and we could solve them we want to solve in order to understand a to get something like a dispersion relationship or something like this we would like to solve these equations analytically of course the problem is immediately these equations are non-linear we are kind of lucky because they are only not really non-linear in this advection term so what we usually do is we linearize these equations around the basic state in order to be able to solve them because non-linear equations cannot very difficult very complex to solve analytically so so what we do is we linearize them a route and we we make it very easy for us we assume that the basic state has zero flow but this probably this assumption could be easily removed by assuming some some some mean flow in the zonal direction some kind of Doppler shift to the solutions that we find but the most easy easy way to solve these equations just to assume no basic flow then linearization is very simple because the non-linear term doesn't give us any additional contribution just vanishes we assume that second order quantities are zero small compared to the first order quantities all the other terms are already linear apart from this equation where we get a get a contribution from the mean height and importantly at the equator we do this also in the mid latitudes sometimes we look for solutions of these equations we assume some kind of approximation of the variation of the Coriolis parameter with latitude so we assume that the the Coriolis parameter can be expressed as a as a constant term plus something that linear depends on the original direction the thing is that this is a Taylor series expansion this term at the equator becomes zero the Coriolis term so what we get is only a linear dependent Coriolis term in our equations no constant term if we solve these equations in the mid latitudes we would these raw equations as they are where no filtering has been yet really applied to then we would we would we could assume in that equation that that we can use here instead of beta y we could use f0 because you can assume with work if we remain close to our base point then the Coriolis falls is mainly due to this mean term in that case we can also solve the equations for the for the mid latitudes we can solve these equations on what the solutions are inertia very fast moving inertia gravity waves with the with the phase speed in this case okay the height of the medium and then plus f0 over k square f square so some kind of gravity waves modified by the by the fact that the earth is rotating so in the mid latitudes these equations I could solve them you know everybody of us could solve them immediately by inserting Fourier components here just no problem immediately you could derive the problem here is it's a it's a slight problem that this is a linear equation we have linearized it but the there are no constant coefficients constant coefficients would make it very easy to solve these equations which is the case for the mid latitudes but for the equator it's not the case there is this this linear varying coefficient of this of this independent variable multiplied with velocity perturbations here so this causes a problem to solve this equation immediately but but of course we try to solve these equations in some ways so and what turns out to be successful I'm I always I'm not good at solving equations anyway but you know you can dream a solution and try if it if it solves your problem so let's say we dream this kind of approach we have an amplitude function that only depends on the original direction and we have waves we assume waves sine waves cosine waves that move in the zonal direction and this this if you if we insert this in these equations it turns out to be a good a good approach because as you as you remember from basic physics if we if we have derivatives applied to this then if we calculate the derivative here with respect to time we get the we get the frequency down and then the sine and cosine they just repeat and if we built the derivative with respect to x we get the wave number the zonal wave number out we have only to continue to use the derivative with respect to this amplitude function if you do this this is very straightforward we get all the for the time derivatives we get all the down from this approach the I knew and then the amplitude function we have divided by the by the e to the power of I and so on to get this this equation for the of our amplitude functions that only depend in on the original direction so again in the mid latitude at this point we would have been the solution would have been there we would have a dispersion relationship the mid-latitude we would have this as our dispersion relationship and we have done we are done here we still have to do some work because we have to solve this for our amplitude functions we are not yet this is not yet solved but you can do some some manipulation you insert one equation in the other and yeah if you do that some remarks here okay if you do that if you insert one in the other here you get the set of two equations here now we try to eliminate between these equations the the mirage in a velocity no the the the height the geopotential height and just get an equation for the but take into account that once we do that we so we divide this by by this factor here and insert this in this equation there we bring this to the other side divide this here we have assumed that this is different from zero okay this something that comes back later so when we're in doing this when eliminating here the the height field we is we are assuming that this here this factor is different from zero and we get this second order differential equation for the original velocity again this equation would be very easy to solve it is in fact the the solution for harmonic oscillator so we come back to we would be immediately back to waves which is the solution that we can derive in the in the mid-legitudes if this factor is was was zero but with this factor this equation again it's an equation with non-constant coefficient so we have we have this problem again and an additional when when when seeking solutions to this mathematicians of course have to take into account that the solutions could should decay away from the equator the the physical reasoning is that when when when doing our approximation f equals f naught plus beta y we we are assuming that that our base point always remains close to the equator so if we go too far away from the equator this is not valid anymore our assumption that our f naught is zero in principle so we have physically we have to to look for solutions that stay close to the equator or put differently decay away if we go away from the equator so therefore these waves that we are looking for almost because of how we we pose our equations are equatorially trapped they have to decay away from the equator so they should be equatorially trapped now and this is something I I have not proven but clever mathematicians have shown that this solution says only the solutions of this are only such that they fulfill this property we can only find solutions that decay away from the equator if this if this factor multiplied by the scaling essentially this factor here fulfills that this is an odd number so 2n plus 1 where n goes from some number so this has to be an odd number and this actually now is our dispersion relationship here we have the dispersion relation now we have a relationship that gives our frequency dependency on wave number and this is what we know as dispersion relationship there's still this parameter n which in principle is still free to vary this n can be can vary and it turns out later that this n is a is the number of nodes we have in the in the original direction of our solutions that eventually decay but before they completely decay they can have some oscillations on the away from the equator so essentially this is our dispersion relation now you do a trick you do a transformation of a variable so you introduce this new variable zeta instead of y and then the the equation looks like this way where we have inserted that this bracket was was equal to this this odd number in principle and again this is often the case okay this equation thank God has been solved already because this is a great this is the equation for the for the harmonic simple harmonic oscillator in known from quantum mechanics for the Schrodinger equation in some ways leads to this this kind of problem so thank God people have solved this equation already and and the the solutions that you find are given by what is called Hammett polynomials so the the solutions as a function of this scaled original coordinate are equal to the to this Hammett polynomial where for each n we can we can find a different one times this this exponential here in fact if you insert this approach in this equation for each n this is the defining relationship for hermit polynomial so if you fall for this hn for the first one for the zero term equal to one insert this here you will find then yeah that that that is the solution and so on so this inserting this here gives us the defining property for for hermit polynomial and the first few ones you can iterate them as many as you as you want the first few ones I have this structure the first one is just a constant so for the for the zero one here where the hermit polynomial is one we get a Gaussian distribution around the equator we get something that decays with a with a squared of the original distance from the equator a Gaussian a Gaussian distribution around the equator however the other ones have some kind of before they decay eventually because this exponential term will always give you if you go far enough a decay is a is a is dominating but there's some there's some polynomials that may change sign in between so we are looking for the case n equal to zero and because here we have to use something that so n equal to zero means that we are looking for very this our original structure is very simple because we are looking for the simple Gaussian decay which may be relevant in reality imagine we have some kind of symmetric heat and so like symmetric heating at the equator then that may force waves that you know have not so much complex original structure but have a very simple structure around the equator and then decay if we go away from the equator also because the heating goes away so I guess it's a physical physical reason why this this most simple case is probably the most relevant one and so we insert this in in our dispersion relationship that we had above and we can do a factorization if you do some some manipulation of this equation we find that there is this factorization however one root here that would give us the phase velocity is equal to this negative gravity wave speed like the westward perfect even gravity wave is not permitted because remember when we divided before we have excluded that this factor was zero that was here when eliminating when eliminating here the the height field we have assumed that this is different from zero therefore the solution is simpler we get just one we just have to require instead of this third order polynomial we got just just get the second order polynomial and the solution is this already it quite complicated from this from what are the ingredients in this square root there is a beta which is the core related to the Coriolis force and there is the the d times h term which is related to the gravity waves so it's kind of it's clear that we have a mixture of of gravity and inertia and rospy waves going on your rospy waves because in fact the beta here is determining also the rospy wave speed you remember the the phase velocities of rospy waves I guess this is all important so the phase velocities of ros of rospy waves in the extra tropics how do they depend on beta and the wave number you remember that for equal beta beta over k square was a highly dispersive are those ways and and it turns out okay there we have so essentially if if if k is goes to zero then this term becomes very dominant and we get some in the end something like inertia gravity waves out of that but if k is is relatively small in the synoptic size you can you can show you can show that this term for synoptic type waves like wave numbers four five three something like that you can show that this is a small term then if you so we have something like one plus x here okay to the square root if you then do a Taylor series expansion you would get something like one half plus minus one half one plus one half of this x value so that would be for beta k square then okay the positive the positive root for the positive root we can essentially then since this we assume that this is a small term this can be ignored and and we get something like gravity wave because this this is determining there that would be the gravity wave speed for the positive root so that would be the eastward propagation for the westward propagation the negative root here this this one half goes out with this one half and what it will be remaining is k e h e that goes also we get one half one fourth four beta over k square so so to a first approximation this goes out this goes out we go back to our exactly to our rosby ways phase speed so for for synoptic type wave numbers our our westward propagating waves at this at this for this simple equatorial solution are approximately again let's do it this way our the same phase speed as our rosby ways in the extra tropics the same dispersion relationship that we can derive for our rosby ways in the extra tropics oh sorry yes yes because this is the frequency so so c is new over k sorry yes yeah so we divide this by the by the wave number and then we get our phase velocity which is the rosby ways so so yeah okay that's that's then interesting so we get we get this rosby waves that propagate with the same phase velocity that we are used from the extra extra tropics we get these rosby ways at the equator equatorially trapped and they move to the to the west just as our rosby ways in the extra tropics one special case is a kelvin wave the kelvin ways are usually the the the prototype for kelvin waves we we find in the oceans kelvin waves are usually something that is that are very important in the ocean where we have borders for example the classical example of a kelvin wave is if you have a bathtub and you you create motion you will find that the maximum amplitude of waves moving around your bathtub is at the border because there's some kind of the border induces convergence and at the border we get the maximum height of our perturbations and the decay towards the inner of your of your of the bathtub and move around that so this border gives us a kelvin wave type behavior in the atmosphere we usually do not have borders but interestingly the equator for the atmosphere at least in this property and for the dispersion relationship as if as if it was a something like a border that's the interesting feature so however there's a trick in order to derive kelvin waves we simply assume and this is the game probably based on observational findings that this is a relevant case if we set the original velocity exactly to zero so for kelvin waves the original velocity component is exactly zero and then we may find different solutions to this to this equation it becomes simple because this drops out this drops out and this drops out so that the solutions become much simpler but we have assumed purely horizontal motion in the kelvin wave so good thing is that the set of equations really becomes now simple so simple that even I would be able to to solve this equation everybody would be able to solve this equation so if we insert if you eliminate now from this set of equations phi between these two so we with the derivative with respect to y here and so this here we get we get this equation here yeah we get this equation here but if we if we derive this far yeah if we solve this for these two then we get a relation for our for our original velocity for our zonal velocity original velocity was zero but zonal velocity has some structure around the equator and this structure we can derive by solving this equation again you can dream a solution I dream a solution if I dream this solution insert this here it can be shown that this is the solution so again we have a gaussian structure around the equator just as before for our simple rosby wave solutions now but now before it was for the original velocity component now for the zonal velocity component we find a gaussian structure around the equator but there's something here because from the dispersion relationship this one here this value c could be positive or negative because we take the square root so it could be plus or minus square root of g times h however clearly if we if we insert the negative c value here then our our perturbations in the zonal velocity would not decay exponentially away from the equator but would increase exponentially as we go away from the equator so we have to include exclude the solution so they can only one for carrying ways they can be only one direction in which they can move and this is eastward so that this has to be the positive root which means eastward wave propagation equatorial rosby waves westward cabin ways eastward and and it's also from this equation it's probably useful to derive the distance from the equator for which the waves become small so let's say one over one over e one almost one-third the e-folding distance with respect to the equator if you just depends on this factor here so if you can work out that the e-folding distance is is this 2c divided by beta square root of this now we have to insert an observed value for c this is a trick here because in theory we know that for this case c equals square root of of the height that we have assumed no one no one scientist can tell us how we should pick this height of our system we use the trick we eliminated and we just use the phase velocity which we can derive from measurements and if we use measurements for the phase velocity we we get a e-folding scale of about 1500 kilometer so the k-wing waves decay quite rapidly if we leave the equator for for the atmospheric case this is of course depends on what problem we are looking at interestingly if you if you do it the other way around you would say so this is 30 meter per second this is a typical phase velocity we can derive for Kelvin waves if we square this we get nine okay let's say 1000 we divide this by 10 so the typical our scale height would be 100 meter I have no idea what the meaning of this height is okay there's a so here's the problem we cannot use any reasonable height in this in this simple set of equations to derive the phase velocity of the Kelvin ways or Rosby ways or any of the ways we have to use observed values this is the caveat we get from over simplifying our system in a way such that okay this this height yeah it's not interesting so it's actually quite intuitive these are the these are two cases for the the upper one is the is the simple it's K-wing wave and here is a Rosby wave for the Kelvin wave it's actually quite intuitively from the motion in this wave to derive the phase velocity it's very simple because these these are the zonal velocities here and these are the this is the height perturbation so negative height perturbation here positive height perturbation there in the velocity field the zonal velocity field is such that it is maximum in these in the also in the centers of the of the height field but look at this what would happen in this in this region here in this region here we have a divergence of our flow divergence of the flow means that we lose mass there that means that this low or height in our simplified case this low pressure that is sitting here will have the tendency to move over here so that gives us an intuitive understanding for the for why Kelvin waves are moving to the east because the the velocity in them is such that the horizontal velocity field is such that it promotes wave propagation in the in this direction okay and then you can of course come back to you can actually draw these dispersion diagrams new with respect to K and for even for different values of n and so on for our Rosby ways and for the Kelvin ways the Kelvin ways in such a diagram are linear give us a linear relationship because because the phase speed does not depend on the on the on the wave number but the Rosby waves are more complicated depending on the on the on the original wave number and on the zonal wave number they they can have different more complicated behavior and of course this remains us this reminds us of our initial diagram so that's the reason now you can interpret this diagram theoretically and say okay this this increased energy that we see here fits to the dispersion relationship the linear dispersion relationship we can derive for Kelvin waves and and it's not it's not so clear actually for the equatorial Rosby ways because the idealized line would follow here but in this spectrum okay it's the real world sometimes can be more complicated but in some ways they are following this this property this face dispersion relationship for equatorial Rosby ways in some ways these waves now will be on Friday I will give a second lecture on on the Enzo phenomenon and the positive atmosphere and ocean feedback and as you can imagine the Kelvin and Rosby wave solutions they play a fundamental fundamental role of the adjustment of the atmosphere to a to Enzo phenomenon in the ocean in the atmosphere but also in the ocean we will we will apply the same kind of solutions that we have applied here for the atmospheric case because we have specified phase velocities that are valid more or less for the atmosphere we will also use them for the ocean and the the the atmosphere and ocean adjustment they work together the wave adjustments work together in order to create a very interesting positive feedback so the the waves that we have derived here are crucial in setting up this equilibrated response because the waves are very quick and then so imagine an Enzo phenomenon will take months to develop if you if you have followed the 2015 El Nino event it has developed already in spring maybe a little bit earlier and is still there if you look at look at daily maps it doesn't change so much the waves that we have been looking at within one day they they've traveled along around the globe so so this Enzo phenomenon is kind of providing a low frequency forcing for these waves and we have to look how these waves equilibrate in order to provide some kind of feedback to this to this Enzo phenomenon and this is what we what we derive in the lecture on Friday this is okay timing okay do you have any questions that someone else then can answer someone understands more