 So ladies and gentlemen, I'm very pleased to introduce you now the next speaker is Professor Jérôme Noir from the Institute of Geophysics of Zurich and he will take us to a trip to the center of the earth. I think also shows us that science is always a little bit of cooking and he speaks on the contributions of Henri Poincaré to the modern geophysics from the internal structure of the earth to the geodynamic. Thank you very much. So first of all I should say that I was very honoured to be invited to this colloc and I think it's interesting because I'm not a mathematician at all. I'm an experimentalist and it's interesting to see that Poincaré had a great influence on people like the community of the dynamo and the dynamic of the core of the planet which is quite far from what could consider as his main point of interest. And first I would like to acknowledge a colleague, David Sebron, who is doing a postdoc in my lab with whom I had an endless discussion about this topic because when Jacques Lascar invited me I thought he would be rather straightforward to put together a talk and the contribution of Poincaré to modern geophysics but rather quickly I realised that it seems so rich that I should have been given a day to go through all the contribution of Poincaré direct and indirect. So I will first present the geophysical context at the end of the 19th century to give you the main motivation of the mathematician at that time. Then I will try to outline how people came with the idea that we could probe the Earth's interior by looking at astronomical observation. Then I will briefly summarise the work of Poincaré of 1910 which is the last of a long series of studies on this topic by many brilliant people and then I will go through basically the 20th century and show you how what Poincaré has done at the end of the 19th century became during mostly the second half of the 20th century and hopefully by the end of this talk I will show you that it is not the end of the story and that we might not know everything about what Poincaré has started. So to start with this is the vision of our planet that we have in 2012. So it doesn't like it. Thank you. It doesn't like the pointer for some reason. I don't know if there is a reason. We're going to take that old screw. It's a heavy one. So basically we live on the crust. We have the atmosphere, the ocean. We live on the crust. We have the Earth's mantle which is a solid on a hundred to thousand timescale but is a very viscous fluid as you go to our geological timescale of million years and about 3,000 kilometres below our feet. You will find a transition from this rock mantle to a liquid part which is called the outer core. It's made mostly out of liquid iron which is a conductive material and some light elements and at the very centre we have a solid almost single crystal of iron that is about 1,300 kilometres of radius and what we know is that these crystallise because of the condition at the centre of the Earth which leads to a release of Latin heat at the interface which we believe is a source or part of the source of the convection that's going on in the liquid part of the Earth and these convective motions are believed by interaction with the magnetic field to participate in what we call the dynamo process which is a self-regenerating process that maintains the Earth's magnetic field. So that's basically our understanding. I didn't talk about the plate tectonics and all the things but that's the structure of the Earth and that's what we think is the fundamental mechanism that produces a magnetic field. However, if we... it doesn't like it at all. Can we get the next one? Excuse me, I'm going to leave it here. Voilà! So... Now I'm going to... So if we go back in time the discovery of the solid inner core is due to Ingeleman in 36 by looking at the propagation of the seismic waves that could be emitted by an earthquake somewhere at the surface of the Earth and by recording the incoming wave at the surface of the Earth one can see that you have a region here that deflects the waves and after some extremely complicated analysis you can determine that this should be a solid of roughly 1,300 km using roughly the same approach so how Dreyfress in 26 definitely refuted the assumption of a purely solid Earth he demonstrated the existence of this liquid core to make it short you have basically two types of waves that propagate inside the Earth you have the pressure wave which corresponds to a compression of the media and you have the shear waves and the shear waves do not propagate inside the liquid and you can easily do this experiment in your bathtub you try to move the surface of the bathtub and you will see that this motion of the surface does not propagate through the bottom on the other hand if you had some tracers like some T particle you could push on the surface and you would see that the wave will propagate so using this kind of arguments Dreyfress and before him Gutenberg basically suggested and Dreyfress definitely demonstrated that the center here is liquid in 1919 it is Larmor who first proposed that the magnetic field of the Earth and actually he was thinking first about the Sun that the magnetic field in planets and stars could be produced by the motion of a conductive liquid in a background magnetic field and finally the first evidence that the center of the Earth might be liquid as I say was due to Gutenberg and so by the time at the end of the 19th century when people were debating about the internal structure of the Earth very little was known and many many different models were outside the first one was a favorite model of the geologist because geologists were most concerned about volcanism and the easiest way to explain volcanism was to have a huge liquid interior and a very thin crust across which fluid could percolate and produce the eruption on the other end of the spectra somebody like William Thompson Lord Kelvin sorry would have considered the Earth to be completely solid because otherwise it was impossible to account for some astronomical observation and then in between we could find a huge variety of models going from a thick shell with a liquid interior to a thin subsurface ocean and even to something that is actually not too far from what we know now a day however there was very little argument to support any of those models actually except maybe for that one and though the question at the end of the 19th century the great challenge in geophysics was can we use some external observation to constrain the internal structures of the Earth or any other planets and the first the first person who proposed this kind of approach is Mr. Hopkins who was a tutor at Cambridge University and he was the first one to propose that a planet with a solid interior and a planet with a liquid interior should respond in different ways when a gravitational torque applied to it and in particular you were thinking about the amplitude of the procession and nutation that would be very different whether you have a liquid or a solid planet you probably, I'm sure you have seen this experiment but to convince yourself that those two objects should behave very differently I have two eggs one is raw and one is a boiled so the boiled one is solid and the raw egg has a liquid interior so if you have a rotating object I'll do it again that's egg number one and that's egg number two I apply about the same amount of force to those two objects and as you can see one would keep rotating forever the other one will very quickly dissipate if I had a bit more space I would have done a more spectacular experiment where you can actually rotate along the long axis but I've been told that the hardwood floor is quite old so I think you can do it at home you would be free to do a mess and what you would realize is that yes indeed a raw egg and a boiled egg can behave very very differently but the same would be true for a planet and I think that's probably the only experiment they had in hand to try to ascertain this statement another interesting thing about Hopkins besides the fact that he proposed that is that he was a tutor at Cambridge and Cambridge has his long tradition that has been lost through time of what we call the Wranglers which are the first and second of each year and Mr. Hopkins has been tutoring about something like 200 students or so and among those people it's interesting to see that you will find Mr. Stokes Mr. Thompson, Alias Lord Kelvin, Maxwell and many others that later on became some of the most famous scientists of the beginning of the 20th century and he was actually known as a Wrangler machine or something like that but he produced a lot of these people and the person who actually really acknowledged him for this idea was Kelvin in 63 and I got that from a paper by Malker that if you have a chance to read it it's extremely interesting where Thompson used to say that from Hopkins to whom he used to use of thus learning the physical condition of the entire from phenomena of rotary motion presented by the surface so now I would like to introduce what we mean by precession and enutation so I think the easiest way to visualize that is imagine you have a gyroscope and so the gyroscope rotates along this axis and you tilt a little bit the axis of the gyroscope because of the local gravity you will have a torque which tends to tilt even more your gyroscope and in response to that torque the gyroscope will start to describe the conical motion around a vertical axis that's what we call the precession and when we look at the Earth the same phenomena will occur the Earth is not spherical it's slightly flattened so you have a little bit excess of mass on the equator and the equator of the Earth is slightly tilted compared to the ecliptic plane since most of the planets lie the orbit lies near to the ecliptic plane at the consequence you have a torque that tries to realign the equator of the Earth with the ecliptic plane and exactly like the gravitational torques try to tilt this gyroscope the Earth will start to precess along a conical surface in about 26,000 years and the difference between those two objects is that in one case we try to realign the axis of rotation with the perpendicular to the ecliptic which produce a precession in a direction opposite to the rotation whereas the gyroscope because of the direction of the torque will start to precess in the same direction now I told you that most of the planets lie in the ecliptic plane not exactly and therefore there is a time variation of this torque that applies to the equator of the Earth and as a result of this time variation you also have small perturbation oscillatory perturbation of the direction of rotation of the mantle on this conical surface and those small departures from the pure precessing motion are called the nutation and so the proposition of Hopkins is to say that if I apply a fixed torque to this object whether it has a liquid interior or a solid interior it would produce different amplitudes of this motion the other very important discovery at the end of the 19th century that motivated a lot of scientists is due to Chandler who in 1991 observed that the rotation axis of the mantle so on the surface basically if you want on the crust the rotation axis that you observe through the stars does not coincide at all time with the symmetry axis that we would call for simplicity is a north-south direction which is the axis of symmetry of the Earth the Earth is almost spherical except it is slightly flattened it is of symmetry of revolution around this north-south axis and what Chandler has observed is that the axis of rotation does not coincide with the axis of inertia and it does actually describe some small circular motion around that this with a period of 433 days which is now known as Chandler wobble on the other hand earlier in 1758 discovered that if you take a solid object a solid object that rotates one of the mode possible solution of this system is a small circular motion with the axis of rotation in other words when I'm saying that if you try to rotate this egg it could rotate around its axis of symmetry or it could rotate like this so the rotation axis is fixed but it is tilted compared to the plane of symmetry of the planet however if you calculate the period of earlier for a solid Earth you will find about 304 days and so people use this argument to say that there might be a liquid region inside the planet that lengthens the period of the free earlier mode of the planet when you look at who worked on that topic it's pretty impressive the beginning was Hopkins fighting with Deloane that's England against France then independently half in the US started to work on this problem and did actually a major piece of work that for some reason is not connected to the other the study of the other people then later on the student of Hopkins Kelvin started to work on this problem while the successor of Deloane was at the National Observatory or the Bureau de la Titude in France I don't know exactly the successor was nobody else and Poincaré and so basically this debate between Hopkins and Deloane continued with a debate between Poincaré and Kelvin and among other people you find Love, Larmor, Rado and Slutsky who was a professor at the University of Moscow who that's a question for you if somebody knows whether Poincaré was aware of the work of Slutsky please let me know because I received the paper of Slutsky only yesterday so I tried to get it but I couldn't find a copy I got it yesterday and there is a strange similarity between the notation of Poincaré that is not fine anywhere else in any of those except in the paper of Slutsky in 1995 so about 15 years before the paper that we will be concerned about whether they believed in an inner core, liquid core or in a solid planet they all found as a conclusion that either the liquid will behave as if it was a solid and therefore there is no way to differentiate between a liquid interior and a solid interior of the notation or the lid is a conclusion that the earth is solid and has a rigidity of steel what's interesting is that we know it's wrong we know there is a liquid we know that from Jeffress 49 to 6253 we know that you can actually have the right amplitude of notation by taking into account the proper elasticity, rigidity of the fluid but what is rather striking is that we still use their model and as far as I'm concerned they remain valid so we did not improve much compared to what these people have done at the end of the 19th century with very little observation no numerical simulation to validate their model and no experiments that's really striking and as I said it's Lord Jeffress who finally validated those models by applying them to realistic notes that's a letter that I found that was sent by Kelvin to Poincare in response of a formal letter of Poincare to Kelvin in 1901 which are quite interesting the first statement of Kelvin which is ever since 1976, so look at the date 1976 when that statement was published I have been looking for time to go through the mathematical work again so I think it's basically legal to reuse that as an excuse for any of us if Kelvin state that you can tell your colleagues that you've been looking for the time to work again on this topic we're talking 76 to 1901 now more interesting is when you go through those letters is to see the respect that these people used to show to each other and how they were doing science at that time they were competitor but first of all they were kind of a generator and I like this statement by Kelvin that say I hope you will publish your own work with your correction to my errors where you could have published something and say I've done a mistake and put at the end of the paper acknowledgement Poincare but that's not what he did he asked Poincare to publish a paper and interestingly enough 10 years later so that gives you an idea of the time scale at that time 10 years later Poincare finally published in memory observation a paper called sur la précession des corps déformables and that's the last paper of a long series by all the people I've cited before and Kelvin is dead in 1910 so this is really the last paper that and Poincare acknowledge that it's just revisiting what people have done before but revisiting in such a way that it becomes probably the most elegant demonstration on some mathematical aspect of this problem and that's probably why people remember this paper and still use it in 2012 so the question at that time was knowing the gravitational torque that is acting in the equatorial bulge of the Earth that we can calculate with some with some uncertainties by looking at the orbits of the planet the question is can we predict the time variation so the mutation and precession and wobbles of the axis of rotation of the mantle which is the only thing we can observe you're sitting at the surface of the Earth the Earth rotates by looking at the stars you can deduce what is the orientation of the axis of rotation of the top of the Earth and the question is if we do a model with a pure solid Earth and a model with a solid mantle with a liquid entire can our prediction be compared to the observation and what can we conclude regarding the internal structure of the Earth I added this transparent because that's not the question that people ask when they reuse the work of Poincaré as you shall see later so here is kind of an it's almost like a summary slash overview of the paper of Poincaré so that's a solid Earth model that would be an Earth with a liquid entire and Poincaré does some assumption he suppose first the mantle to be rigid then it will reintroduce elasticity then for the fluid inside this cavity he will assume it's non-viscous and incompressible which for a fluid dynamicist is the first thing that you're going to do you don't want to have to deal with viscosity and you don't want to have to deal with compressibility but at that time I think they had absolutely no argument to make this kind of assumption but it proved to be correct and finally he will first assume that the core mantle boundary here and the mantle itself in both cases had the shape of an ellipsoid with three different axes A, B and C and very quickly in this paper he will he will suppose that it's an object of symmetry around the rotation axes by stating that A is equal to B different from C and he will apply some fundamental conservation laws such as the angular momentum the angular momentum is similar to the momentum of an object that moves linearly an object that moves linearly you can say that the mass of these objects times the velocity which is the momentum varies in time depending on the force that are applied to this system the momentum equation is slightly different what replace the velocity is the rotation rate and what replace the mass is the inertia and the inertia it is an information about the mass and how the mass is organized inside the solid and so the variation of this quantity the angular momentum must equate the torque that you apply to it so if you don't apply a torque you conserve the angular momentum if you start to apply a torque there will be a response of the system according to this equation and that's basically the gyroscope effect I was talking about that you also get if you rotate a bicycle wheel and you try to tilt it you will see that this torque that you apply produce a rotation a time variation of this angular momentum so you have two models in this model you have two layers if you have two layers the total angular momentum of this planet has a contribution from the liquid part in our core and a contribution from the mantle you cannot observe this but you can observe that so the first step in the work of Poincare is to see how can we determine the motion inside this liquid layer and how can we express omega as a function of capital omega the one of the mantle if you can do that then in principle by ranging in this equation you can write down an equation only for capital omega and if you can solve it then you can compare your prediction to the observation the beauty of the approach of Poincare is to say and I think that's what you said about change of coordinate system I know if he was that has been suggested this transformation of coordinate system has been somehow applied by Brian a couple of years before but Poincare doesn't mention him so I don't know if he came to the same my guess is that he developed this kind of approach before and Brian got inspired and basically Poincare reused his own approach extremely elegant you have actually leaps through it and believe me at the moment I'm trying to solve this equation directly to this object it's terrible and so Poincare said okay to each particle of the fluid in this object I can associate an artificial particle in a spherical domain by simply applying this geometrical transformation and since the velocity of the same particle is simply the time derivative of its position you can show that the velocity of a particle here translating to the velocity of the artificial particle in the sphere as so and now what he did is to solve this problem in this spherical domain with as a constraint that the fluid that moves inside the sphere cannot go out so that's a rigid body to represent your calm boundary so he did that with the assumption that the fluid is inviscid and something very important that I forgot to mention earlier he does an assumption on the nature of the flow that will be driven inside this liquid layer he says and he acknowledged that this has been that's where he's revisiting the work of Kelvin he knew a little bit about what was going on and he said that I'm going to look for the simple motion possible which is time independent and linear in the spatial coordinate and Slutsky that I was talking about instead of stating that says that I'm going to look for a velocity field that derive from a potential which is essentially saying the same thing so that's the assumption and what we're going to show is that if you have the mantle of the earth rotating along the blue axis which precess along the red axis so it moves on this conical surface the fluid inside the core takes a form of a solid body rotation so a global motion of all the particles plus a small component that we call the gradient flow simply because when you move from the sphere to the ellipse you still need to ensure that the fluid is not leaving the core that you have a solid boundary and that's an extremely elegant way to present it and in addition he now injected that basically in the fundamental equation of the hydrodynamic to determine how omega and the rotation of the mantle are related to each other due mostly only in that case to the pressure torque that comes from the ellipsoidal boundary and what he found that the direction of rotation of the core is fixed compared to the rotation of the mantle and the precession so all those three vectors first they are in the same plane and they are fixed relative to each other and the liquid rotates around an axis that is slightly tilted from the mantle very slightly so that at first order one could actually approximate this motion by a rotation along the blue axis plus a small component in a direction perpendicular to the axis of rotation of the mantle once he determined how the how the rotation of the liquid core can be expressed as a function of the rotation of the mantle its time derivative and the geometry of the system if in addition you you express the moment of inertia which depends only on the type of matter also the density the geometry of the system and of course the thickness of the mantle that you are considering and if and if you can't express how the time derivative of the angular moment of the inertia of the mantle depends on its rigidity so in other words its capacity to be deformed then in principle now you can inject all that in your angular momentum equation and solving for omega you can derive an expression for omega as a function of time or d omega as a function of time that you can then compare to your observation and so that's it's a very sketchy summary of the work of Poincaré but as far as I am concerned it's sufficient to ascertain his contribution during the 20th century so I'm not going to go into the mathematical details of the derivation so the first and most important statement is that the fluid in the core essentially rotates along an axis that is only slightly different from that of the mantle then the next conclusion which is quite obvious is that if the core mantle boundary is spherical the fluid because its inviscid is completely decapable from the mantle and the system will act as if only the crust was subject to this gravitational torque and because of the low inertia of the upper part of the planet you should expect huge amplitude of mutation which is not what is observed so one can reject a thin crust model with a spherical boundary if the boundary is spherical what Poincaré found is that for very low frequency perturbation so long period perturbation such as the precession for instance the interior really behave as a pure solid and the solid I mean this liquid that move as a whole almost like the mantle it's exactly the same as saying that the planet is completely solid you can't really tell the difference between the two however if you apply a higher frequency perturbation in the gravitational torque then a liquid that is rapidly rotating can exhibit free oscillation and this free oscillation together with the natural free oscillation of a rotating solid body will lead to some particular frequency for which depending on the actual internal structure for which you could get a resonance in the system so just to I'm going to try to do an experiment to show you that indeed a rapidly rotating fluid exactly like the string of a guitar or of a laser cavity can also exhibit free oscillatory modes so a typical experiment for that is to you take water you put T particle the T particle fall at the bottom and now you put this system in rotation are you going to have to be very quick because the viscous effect will damp the phenomena that I'm trying to show but if you look at the center you will see that the T particle collapse at the center but you will see some form of a breathing of the system let's see if we can observe it if you look at the center you see that they spiral to the center then they go outside spiral to the center then go outside and so on with a frequency that if you look carefully is typically of the order of the frequency of rotation of the fluid inside there is another striking thing here which is that the T particles that you see are sitting at the bottom if they sit at the bottom that means they are denser than the fluid if they are denser than the fluid as you rotate they should be centrifuge out and in fact what you see is that they collapse at the center and that's an effect that I would have liked to avoid which is the effect of the viscosity in a rapidly rotating fluid which is very different from what you would observe if this system was not in rotation so that's what we call an effect of what we now call the Eckman pumping so I don't know if you've seen that you really have to know what you are looking for if the container was twice the size it would be obvious but I think you can see a little bit of that at the center that tends to collapse and then they get moved out and so on and that's one of those that's one of those free mode and a free mode if you were to excite this system at the frequency that corresponds to this free mode it will resonate and its amplitude will become extremely large it will grow exponentially the same thing applied to this and this planet if you excite the system at the frequency of one of the free modes you should expect a resonance and it's not clear to me what point can I conclude from that because at the very beginning of his paper he says in principle I agree with Kelvin that say that the Earth is either solid or we cannot make the difference and at the end it tells you that depending on the structure of the planet there could be some oscillation that would lead to resonance but you don't see any resonance or you don't see any very large mutation amplitude so either you conclude that it's a total it's a rigid planet or for which you get a resonance are not that of the Earth so Poincaré died in 1912 and unfortunately about a year before Gutenberg presented the first suggestion that there might be a liquid core inside the Earth I don't know whether Poincaré was aware of the discovery of Gutenberg my guess is yes but I couldn't find any letters that they exchanged the next major piece of work on that topic due to Harold Jeffress who applied those models to a realistic Earth Harold Jeffress was a precursor of seismology and so he had the right rigidity and elasticity of the mantle which allowed him to derive the right the right value of the mutation amplitude and over the 20th century the major question has shifted toward the following what creates a magnetic field of the Earth what is the source of motion that produce a magnetic field of the Earth and how much dissipation would be associated to the motion inside the liquid of the Earth in other words how much heat could be produced and that is very important if you want to build a thermal history in all of the planet to ascertain its age basically so the question that we were asking we as the community is slightly different we know the size and shape of the liquid core within some reasonable uncertainties and we know the drift of the mantle so we know the motion of the axis of rotation of the mantle from astronomical observation so that is no longer an output it's an input of the problem and now the question that we are asking is what is this motion of the mantle producing inside the liquid core if you try to solve this problem you very quickly find something that is difficult to identify in the work of Poincare which is that if the fluid is inviscid and the cavity is a symmetry of revolution there is actually an infinite number of solutions for the rotation of the fluid and so if this fluid moves as a solid and rotates you don't actually know this speed of rotation however in a planetary application because the rotation is only slightly different from that of the mantle I told you you can do the assumption that along the rotation of axis of the mantle the core rotates at the same speed and develops a small equatorial component so if you add this condition another condition then you do a unique solution so what Poincare tells us is that if you ask me that question I will tell you that the fluid mostly rotates around an axis slightly different and by applying this ad hoc condition I can give you the direction and rate of rotation now there are two remaining questions one is what is the effect of the viscosity of the liquid and the second is the solid body rotation is that stable or can that collapse to some turbulence turbulence means dissipation and so those two questions are related to the two problems I told you about with the third one which is that unfortunately I'm not sure Poincare would have been very happy about that but after the second world where the geopolitical situation was such that people started to consider sending rockets from one end to the world to the other side of the world and you stabilize the subject by rapidly rotating them around their long axis and as they travel around the Earth they basically precess and so if you use a liquid fuel then you have the same question as the people who care about the thermal history of the planet to some extent as far as the viscosity of the fluid is concerned there are a couple of contributions through the 20th century I'm going to focus on the last one by Fritz Busse in 68 who basically assumes small ellipticity small viscous effect and start with the inviscid form of the solution of Poincare so he will start with that at the equation of hydrodynamics including the viscous friction and find that in that case there is a unique solution to the equation so you don't need this ad hoc condition it naturally appears in the equation that it's replaced by this solvability condition which it's only slightly different and I'm not I don't want to comment about it what I've done is in the last few weeks I've carried out some numerical simulation in a spheroid basically to ascertain the efficiency of the Busse model and how it compares to the Poincare model and so Poincare is in red it's inviscid so you have this resonance and I might go back to that resonance later that is in gray Busse and what you see is that for a moderate so significant effect of the viscosity in a small ellipticity regime yes Busse does a good work in maximum amplitude near the resonance while Poincare has the amplitude wrong but it's almost correct at first order when you increase the ellipticity however Busse which does this assumption fail to reproduce your numerical simulation whereas Poincare which does a very simple and drastic approximation does reproduce quite well the frequency at which you expect the resonance and outside of the resonance even the amplitude of the flow when you apply those resonance to the Earth so you have about 45 pages of equation to end up with this estimate of the equatorial component of rotation and here you have about 20 lines of equation to end up with this solution so I let you appreciate the progress that we've made I'm a big fan of Fritz Busse he was in my PhD jury I like the guy but it's a lot of work for the reader to understand something that only produce such a small deviation from this elegant solution of Poincare now to ascertain the stability of the flow that's something that Poincare could not even consider at that time to use experiments because numerically it's extremely difficult especially 20 years ago and so people started to build experiment this is the one I built during my PhD it's not that I want to show you what I do it's just that all the people who worked never published a picture of their experiment which is a shame but they're all built in the same way so you have a container on the outside but elliptical on the inside that rotates rapidly around an axis and you impose a small rotation of this turntable that will force the precision so here you have to be careful that this system has an infinite source of momentum through the torque applied by the motor to the bottom table and there was a series of experiments that were conducted between the early 60s until basically this year the pioneer work is due to Malkus who doing such an experiment in a slightly ellipsoidal container showed that the fluid for low precision rate behaves almost as a solid with some structure inside that remains organized along the axis of rotation of the tank and as you increase the rate of precision so the rate of rotation of the turntable you end up with a system that will exhibit turbulence and turbulence must be associated to small scales and therefore dissipation however Malkus did not actually look at the solid body response of the fluid which is truly what Poincare is giving you and so during my PhD the first thing I did was to see whether there was a solid body solution in this system so by putting some small light elements inside you can visualize the axis of low pressure which is your axis of rotation and in fact as long as the precision is small it is slightly tilted compared to the axis of rotation of the container that is here as you start to increase this you find a resonance of the solid body rotation mode which appears to be exactly a free mode of the system and what you see that as you path through this resonance then you lose your axis of rotation and slowly you move toward a system that exhibit less and less rotation and more and more turbulence which again suggests that I have dissipation and complexion of motion to produce a magnetic field the underlying mechanism of the stability of the Poincare flow is due to Richard Kersuel in 93 who discovered that in an ellipsoid if you have a rotation that is slightly tilted you have two effect one effect is that if you look at the streamline of the rotation because of the ellipticity when you want to tilt the rotation you have to shear the streamline so by tilting the axis of rotation of the fluid you induce a shear of the streamline and because you now no longer rotates around the symmetry axis of the container you have elliptical streamline and what Kersuel showed is that both of those effect can couple together three modes of a rapidly rotating cavity so you have two free modes that are coupled together one of those phenomena leading to exponential growth of the kinetic energy which ultimately turn into turbulence and so the question was are those instability what we call dynamogen are they capable of produce a dynamo so a dynamo process in few words it's simply it was proposed by Larmor is that you have motion negative liquid so kinetic energy input in the presence of an existing magnetic field by the Faraday's law it tells you that you're going to produce electrical current and the electrical currents by the Emperor's law will produce an induced magnetic field and if the induced magnetic field is parallel to the imposed magnetic field then you basically reinforce the original magnetic field that would otherwise disappear and in the case of the Earth it would disappear in 15 15,000 years whereas we know that we have a magnetic field of about 4 billion years and so the key question now is does the motion have the right topology to produce an induced field that is parallel to the original field and so Malkius based on those observation suggested that maybe the precession of the Earth but of other planets could drive the dynamo and I don't want to get into the detail but there was a debate it was more than a debate actually it was a fight, a true fight regarding do we have enough kinetic energy and it's a in 96 who finally put an end to that discussion but if you don't have the correct topology of the velocity field then it won't be a regenerative process you will create a magnetic field that may be a 90 degree from the imposed one in which case you have no chance to build up a self sustained magnetic field the actual proof that the flow driven by precession could indeed produce a dynamo is due to Andreas Tigner in 2005 who using numerical simulation showed that what he called the laminar flow which is nothing else and the solution proposed by Poincareer 100 years before can indeed produce a dynamo however this mechanism because it rely on strong viscous effect is certainly not dominant for a planet and he showed that the instability that have been observed in the experiment can indeed drive a dynamo process and yes and even more important that it is likely to happen in planet like the Earth I'm not saying that the dynamo of the Earth is due to the precession, don't get me wrong I don't want to get into trouble but for some planets that we know for sure that cannot convict it is a possibility and I think that it's not a tribute to Poincareer it gives you an idea of how far how deep inside our community the influence of Poincareer has been because all those research that have been conducted on that topic leads to the idea that the precession which was clearly first studied by Poincareer could produce a dynamo and I have colleagues in Dresden that are creating this totally crazy object which is a giant gyroscope of 1 meter by 2 meter filled with a very good conductor which is liquid sodium which is extremely dangerous and they are going to preset this object so it's very similar to the experiment I showed you except for the torque and the field that is of about 1 million Newton meter but aside from that it's the same object and what they aim is to demonstrate that experimentally that the precession can indeed have a homogenous dynamo in a similar way to what you can find in the Earth's core so basically Poincareer and others it must be acknowledged that there are many contributors to this problem they have developed a framework that has been used over the 20th century by many geophysicists to ascertain the question of the core inside, sorry the flow inside the core of the planet and the model presented in 1910 and earlier are still valid as far as I am concerned if you do not ask the question what is the unstable flow in the system if you ask the question where is the core rotating those models are perfectly valid as I was preparing this talk if I can have two more minutes is that okay? very, very quickly I was preparing this talk and some colleagues of mine have produced a paper regarding the moon and the possibility that the early magnetic field of the moon would be due to the flow driven by the precession at that time however the moon is not an axisymmetric planet the moon has B, B and A so it's more like a rugby ball and I was a bit skeptical that you could just apply Poincare as they did in such an object and the main reason is that if you remember what I told you what Poincare is telling you is that the planet rotates around the blue axis and the core would rotate around the green axis which is essentially the blue plus a small component here and all those are fixed in space so if you were to look in that direction over time this object moves around so if you look at the cross section here what you see is a circular shape then an ellipse then a circular shape, then an ellipse what that tells you is that the moment of inertia in that direction is changing in time moment of inertia, high moment of inertia low, high in a similar way to an ice skater and an ice skater is spinning fast when the moment of inertia is small spinning slow when the moment of inertia is large in virtue of conservation of angular momentum which we might not have in this system we might not have conservation of the angular momentum but it looks similar enough so that you can ask yourself whether the solution of the system will be stationary as claimed by Poincare or not and if you do a numerical simulation in such a system the first thing that you see is that no, the the direction of rotation of the liquid core is not stationary and so there is some form of a paradox here and you do see that you have low rotation when you have high inertia when you have low inertia and I think what is interesting here is that we are now about 102 years after the publication of Poincare and this problem has not been looked at before and it seems to challenge I say it seems it seems to challenge the fundamental hypothesis of Poincare that the flow is stationary and I will end with that because I don't know the answer to this problem Thank you very much Professor Noir for this beautiful lecture which I think reveals maybe less known aspects of Poincare's contribution There is time for maybe one more question Sosinai What is the quarter of magnitude of the distance between the axis of rotation and the axis in ellipsoid of inertia how much a couple of meters it's not kilometers it's very small very small and the second question can you comment a little bit on the models which describe the dynamics of the axis of rotation as a random process governed by some number of fundamental parameters you mean the fact that you don't have a unique solution you have a unique solution but it has random force which produces the motion random force which depends on the motion of atmosphere around the Earth the distance of such small scale then the wind for example in Africa can just have the influence to the trajectory of this axis Okay, so I guess what you're talking about is the Schamler wobble and the effect of the atmosphere and ocean on this apparent motion of the axis there is indeed a very interesting paper that was published that shows that the lengthening of the of the earlier period due to the core is not dominant in this system so that the effect of the ocean of the atmosphere is crucial to reproduce correctly the observation of Schamler and at the core I think we looked at that with Nikolai what 50 days 50 days is the contribution of the core to these 100 days but what is interesting is that if you take into account the elasticity you would go way over the period of Schamler and it's the additional effect of the ocean and the atmosphere that will bring it back and then the core and so it's a rather complicated series of contribution okay I don't call that random but I see what you mean now by random and none of those were known to any of the scientists that work on that problem we have to remember that the absolute beauty of this work is that it's valid it's been used in addition to modelisation of the atmosphere and ocean but these people had no way to validate or even test the model and so when you look at the complexity of how the atmosphere and the ocean contribute it's not surprising that they couldn't use this information to constrain the presence of the core I have a small question I think we should leave the discussion but just I want to thank Jérôme for the way he took his assignment and making it so working so hard that he even made a publication out of it soon soon soon but I had a question about the inner core in that in Poincaré flow how does it contribute for an inner core of the size of the Earth it's a small contribution because well it's not clear I will not give you a personal argument but the argument of Andreas Tigner that is to show that and it's correct that if you do numerical simulation what you see is that it has very little influence on the resonance of frequency so at which frequency you might get a resonance between the precession or the nutation and this mode in terms of the amplitude of the rotation so the kinetic energy it seems to be simply related to the ratio of volume of the liquid core minus the inner core there is no evidence that it will drastically change your view of the problem as long as the ellipticity remains small so thank you very much you're welcome