 So, first of all, I am very grateful to Professor Mahajan to provide this opportunity to share my calculations and results with the experts and also some students. So what I will do is that I will present my calculations as well. But introductory part is also there about the problem which I am doing. It is not directly related with fusion, but it is about hot plasmas and general plasma dynamics. So, our aim is, I will go to the next slide from here. Okay, the aim is to understand the physical mechanism for the generation of seed magnetic field. And then to find out two-dimensional and three-dimensional analytical solutions of two fluid, MHD and neutral fluids equations with given spatial, I mean, the structure on space is assumed to be given by density and temperatures. And then using justifiable assumptions because we are trying to do analytical calculations so we need some assumptions. And to search for a possible mechanism for the generation of vertical flows. I will introduce this problem later. So how the vertical flows are produced in the plasma and to investigate the formation of augured structures such as speakeals in the solar atmosphere. Yes. Okay. So first the observations. The observations show that all galaxies almost have magnetic field large scale magnetic fields of the order of 10 to minus six gauze micro gauze and lower limit on the seed magnetic field is that it should be greater than 10 to minus 25 21 gauze. Therefore people have tried to obtain the seed magnetic field by using plasma dynamics, which must be greater than 10 to minus 21 gauze. And other observations tell us that the material outflows from the plasma in astrophysical environments are very commonly very common and have been observed in many objects. And the first observation is long ago by Curtis in 1918. And there are many other I am just quoting a few just to give you the flavor if you want to consult the literature. And then there are, there is huge amount of literature about sun where the speakeals. Coronal loops, prominences, flares, coronal mass ejections are observed in sun's atmosphere and solar wind also is manifestation of the particle outflow from the plasma. So these are just a pictorial view of the galactic magnetic fields which are on parsec size, large magnetic field scales magnetic field magnitude are small, but scales are very large. In initial laser plasma experiments which were performed with the low power low intensity lasers and in those days in early 70s. The laser intensity was 10 to 12 watt per centimeter square approximately and laser pulse time was in nanoseconds. So the laser hits here, the target solid target. This red is the plasma density gradient is electron density gradient because plasma increases in opposite direction. And if this is the temperature gradient. So the density gradient and temperature gradient here curl of the vector product of density gradient electron density and temperature gradient produces time dependent magnetic field. So this was the basic idea. And these magnetic fields have been observed in initial laser experiments, then in accretion discs the relativistic jets are produced of electron positrons. And they have been observed large, they have large velocities. And this is the jets observed in young stellar objects. Here the plasma is classical and speeds are relatively slow. So now we just discuss to model the plasma outflow and the generation of mechanism for the generation of magnetic field in a simple way, considering the set of two fluid plasma equations. So this these are the electron equations momentum conservation and then I mean density conservation or mass conservation. And similarly these are the equations for ions. And this is equation of state for ideal gas law. And since these equations, these are charged particles. So these fluid equations are coupled with the set of Maxwell's equations. Our aim is to find out analytical solutions of these coupled nonlinear partial differential equations and highlight the physical significance of these solutions briefly with some possible applications. And before doing that, I want to introduce maybe many of you know it very well, but the beer man battery concept. The beer man was trying to explain how the seed magnetic fields are generated in these stars. So in 1950 he presented the idea that if there is temperature gradient and density gradient in electron fluid. And he assumed that this magnetic field is produced on a longer time scale, compared to electron plasma oscillation. So this assumption that always much greater than the time period of electron oscillation, he ignored, we can ignore the electron inertia me tends to zero. And using other part of the assumption that always much smaller than iron plasma oscillation time. He assumed that we can ignore the iron dynamics so ions can be assumed to be static electrons are inertial as so a very narrow window for this. And this term, Lawrence term V e cross B was also neglected just to start with a very simple physical model. So in this under these assumptions, the electron equation of motion becomes very simple equation time electric field balances the pressure electron pressure gradient. If we take curl of this equation. We see that magnetic field is produced, if del n e and del t are independent of time, so then it is very easy to integrate. If we integrate this equation. We obtain this equation 12 which tells that at any time tau later after t zero, the magnitude of magnetic field will be. So del n e divided by any can be written as log any, and that is the cross with delta t. Now application to galactic plasmas. So this theoretical model has been applied to estimate the seed magnetic field generated in galaxies. So, actually, I have said that it is assumed that I am in the background station. This is the model beer man's model, we will see later. So, in this publication, he assumed that the thickness of the cloud because they divide galaxy in columns of different columns of gases large scale. And for example if we consider a column of gas with thickness 10 to two parsec and parsec is you know very large I have mentioned somewhere 10 to 13 kilometers three 10 to 13 kilometers. So if thickness is 10 to two parsec, and he assumed galactic radius 10 to four parsec, then using that equation beer man's battery equation, the magnetic field after time tau will be expressed as equation 30. Now, del n e over any has been estimated as exponential here one over edge. That is the thickness of the cloud and temperature gradient he estimated from edge of the galaxy to the center and he took it 10 to six. 10 to six kelvin and tau that time of galactic is magnetic field generation seed magnetic field generation is approximately the time of formation of galaxies and galaxies are assumed to be formed in first billion years after the creation of the universe. So, during this time, the magnitude of magnetic field turns out to be 10 to minus 17 cows, which is greater than that so that was the seed magnetic field, as Professor Mahajan yesterday was telling that that is then amplified by the dynamite effect to present of micro gauze. Then the same theoretical model was applied to explain the generation of magnetic field in laser plasmas classical laser plasmas. So then that equation can be written in this form as in equation 14. So, assuming that the density gradient temperature gradients have these order, and in those times, the plasma produced had the electron density is 10 to 20 per cubic centimeter, and temperature one KV CS the sound speed turns out to be three into 10 to seven per second, while omega PI is 10 to 13. So, putting all these values, the magnetic field turns out to be 0.6 10 to six cows, and it was completely in agreement with the observations mega Gauss fields were observed in those days in laser plasmas. So computer simulations and other calculations take into account different effects of ions, plasma rotation resistivity many things with simulation. The model, which takes into we all know that there is m hd we have heard a lot of about m hd to fluid model or multi fluid if you have dust then you can increase. And if temperatures are important and micro instabilities and temperature effects appear, then we talk, or we consider the kinetic model kinetic theory. There, and the model, which works in this time scale, where we ignore electron inertia and assume ions to be static is a very narrow window of time. And in this time, we see that electron inertia is ignored ions are assumed to be static. Now, if you look at the laser experiment, you can see that omega PI is 10 to minus 13, and tau, which they assume was about 10 to minus 10, much longer than the ion plasma oscillation time. So that was the initial point from where I started looking at this problem that this is inconsistent to assume ions to be static. But the problem was how to take into account ion dynamics because then the system becomes very complicated. And I don't know computer simulation so we need to formulate problem to do analytically. So these were some comments on beer man battery. First is the as I have shown the inconsistency with the consistency with the application and the theoretical model. The problem is that this Lawrence term B cross V e, as magnetic field is produced as soon as then this term should be operative in the system electron dynamics, and when we apply this model to galaxies for one billion year assuming the town time, and we assume that ions are static. This seems to be maybe for initial calculations it is okay, but one needs to consider all these things. So about emhd, I have written few my comments and calculations, very long time ago, but now in the, and this is the, for me, useful point because most of the work has been done in isolation. So here this is an opportunity to discuss with you people. As I told you that numerical simulations can take into account many effects, which analytically we cannot. But our interest is to find out analytical solutions of plasma equations, which can explain the generation of seed field magnetic field, along with flows, because we want to take into account electron flow flow and flow. And this is just symbolic. This is it means ions can be assumed to be stationary. I am presenting. Yes, I am just presenting beer man battery effect. What is this, and its applications. Now we will. We will discuss on dynamics also I am going to discuss that. Okay, so important point to note is that the magnitude of seed magnetic field produced by density and temperature gradient is very small this we will see in calculations. But the flow produce the vorticity and it is directly related with flow that turns out to be very large. So this is also a point I think the baroque linear vectors, these gradients, they produce large flows in plasma. So two dimensional, actually I started this work when I was visiting Japan, and I write a paper with Professor Yoshida. So in this theoretical model, two dimensional solution. There was one problem which I also noticed later when I understood this problem more deeply that one of the component of the magnetic field. For example, we have three dimensional magnetic field. One of the component remains constant. It means that system has already some magnetic field, and we want to generate magnetic field, so there should not be any magnetic field given. So what I did is that using the same assumptions that cosine neutrality because we are considering large scale dynamics so cosine neutral plasma temperature gradients are there. Now J denotes electron and ion so we will now consider both dynamics and density gradient is also there. So this is produced with these gradients, and they are non parallel, because we want to take curl of these del n cross del t so if they are parallel it will become zero, the mechanism will disappear. So flows, this is the restriction which I impose to simplify the model that long flows are longitudinally uniform said divergence vj divergence of both flows is zero. So we assume that density is constant in time as in beer man battery, but we know that such a system cannot remain independent of time density will also evolve the time, but that is a later problem. Other dimensions and assumptions are same be ignore electron inertia. So in this case, the electron equation of motion becomes 15 equation 15. Here we can be related with iron velocity and curl of B through ampere's law, because when we ignore the displacement current Maxwell's equation reduces to ampere's law and then these velocities are related. So replacing this V e, and you see the curl of equation 15 becomes in this form and here left hand side is the generation of magnetic field on right hand side. The second term is the same as beer man hat, but this term which is non linear v cross B. This is problematic complicated term. And if we replace V e by ampere's law, it becomes in the form of 16. So here we see that left hand side has the time dependent magnetic field. So if non linear terms, one, two, and three, which are very complex complicated. If they vanish, we have the same form as beer man hat, considering the electron equations. And if we look at Einstein's equation and take curl of this equation, it becomes in the form of 17. And again it has these non linear terms very complicated terms. And we have used this classical gas law. Now we look at continuity equation. So if density partial T and is zero as we have assumed that density is time independent, and then flow is longitudinally uniform. And we have an other condition on our flow and density gradient that del n dot V j must be zero. And if we divide with n, we write instead of density, we write psi. So del psi dot V j should be zero. And ampere's law also requires that the gradient of psi dot del cross B must be zero. So these two conditions we need if density is time independent and flow is longitudinally uniform. In addition to these conditions. We also assume that all complicated non linear terms vanish with our solution, which is, which has to be proved. But in the beginning it looks that we have dropped but not we have not dropped. We want to find out structure of psi and temperature gradient in such a way that all these non linear terms automatically vanish. So this we will see later. So we have two equations equation 20. This is just beer man battery and equation 21 tells us that magnetic field is not independent of vorticity. And ions vorticity is linked with the baroclinic vector of ions. And if we add these two term equations are we subtract 20 from 21, we see that the addition of baroclinic vectors of electrons and ions produces ions vorticity, and which is linked with electron velocity as well through ampere's law that we know. So this equation 22 is also an important result. Since we see that vorticity, I am vorticity is linked with magnetic field it can be parallel or anti parallel to the vorticity. So let us assume that magnetic field is directly related with the ions vorticity equation 23. So now we have to choose the form of psi temperature spatial profiles because these are given. And in addition to this, we need to satisfy all the conditions which we have mentioned. First, I just want to see that how non linear terms can vanish. So suppose velocity has this structure that curl of this non linear term vanishes. Okay, we will prove it later but suppose that this vanishes. And this velocity structure has the form that if we take del square, we express it as eta vi and eta is some constant. So these are again two conditions on the velocity. And we will find and see that velocity turns out to be in the form of psi. Therefore, psi should also satisfy the similar equation as 26. So, 24, 25 and 26, along with these conditions that divergence vi is zero, B is proportional to the ions vorticity divergence psi dot vj is zero and divergence psi dot del cross B zero. If these conditions satisfy, then we find a very neat and clean two equations, which produces vorticity and magnetic field. Now I just show that how these non linear terms vanish. If this we have shown that if curl of vi cross del cross vi is zero, then this term del cross vi will be zero because V is the curl of vi. So if this is zero, then del cross vi cross B is zero. Now we look at another equation. That is the vorticity equation we related here. So take curl of this equation 23. Curl of 23 gives del cross B in terms of velocity, and we see the divergence vi is our assumption that it is zero. It means del and del square vi is eta vi. So now we have found one thing in our system that velocity is curl of magnetic field and magnetic field is curl of velocity. So these are the characteristics of our flow and velocity fields. Now this is again a third term. Here it is ions. It is automatically related with the electrons. So yeah, this is the model which I am it is modifying this beer man battery effect from the beginning. That was the other part. This is the second part. So we are taking into account on dynamics. Okay, so this term if we write it becomes in the form of vi cross B curl of it and we have seen in equation 27 that it is zero. So this term also finishes. Now we look the last term that is del psi cross del cross B del cross B. So we just write it in this form and it turns out to be in the form of equation 30. So divergence psi dot B B is equal to del psi cross del T. So this dot product vanishes and this from continuity equation we have seen that they should vanish. So interestingly on all nonlinear terms can vanish if the form of velocity psi and T we found appropriately and that is the main step now that how we can. It is not linear process what is complicated process, but I am trying to simplify the problem to find out analytical solutions but of course, this will just give an insight to the physical processes and I mean, basic mechanism, but people can now try if we put for the physical linear terms, I cannot solve analytically. And also these nonlinear terms I cannot solve. So, okay, I think later on we can discuss but here as an exercise, I am presenting three dimensional solution just a mathematical exercise, I do not have found any application of this but then I will show you the two dimensional solution, it has very interesting in my opinion, very interesting application that we will discuss. So quickly I just tell you one thing you can check because but this is some cumbersome. Three dimensional solution requires a lot of algebra to prove, but two dimensional is very easy. So, this form of psi, I don't know how physical it is or not. So three dimensional solution we just see as mathematical exercise at present, I am trying to find out its application but later. So if psi has this form and T has this form, you see if we take gradient of T, it becomes just constant because this T not j prime is the gradient of temperature which we assume is a constant. So then what happens is this term del psi del Tj because these are the source terms. It has the form of psi because T not j prime comes out of the equation. Cross product and all terms are in the form of psi. So, V i turns out to be in this form equation 34 in three dimension and magnetic field turns out to be in the form of 35. So this is the mathematical expression, its application I am not presenting here and I do not know at present. Now we discuss. Okay, to neutral fluids because it is a simpler set of equations to many equations. So we have more general solution. So here the solution is psi e raised to power alpha plus beta plus gamma. So if I write psi in this form and temperature remains laying linear as earlier, we have again the vorticity produced by this vector and psi has this form and velocity that far. So now I discuss. And that can also be applied to m h d I don't know I have okay so single fluid also but now we do slightly interesting problem with application that is the two dimensional solution which I will present just after this. First we make some background general what we are doing. So if we look at sun sun is divided into two main regions exterior and interior interior has core where the huge thermonuclear reactor exists and energy is produced here. Beyond core is the radiative zone here the gas is almost static and absorption and emission processes are taking place. And then comes the convective zone where the plasma is in motion above convective zone is photosphere. And above chromosphere is Corona even our earth lies in Corona but we will study and discuss Corona closer to the chromosphere just upper. So this is the figure I took from priest book, you can see it. So I just go here to give you some flavor of solar structure for the application so that you can appreciate application. The thickness of the photosphere is approximately 500 kilometer and density is 10 to 13 electron density and temperatures are low. Therefore, neutrals also exist there. And this is the reason that we have visible light mostly visible light from the sun surface and the spectrum of sun although it contains x rays ultraviolet etc. But most of the part and intensity comes in the visible spectrum. The gamma rays which are produced in the core when they pass the radiative zone and emission and absorption processes takes place. And they reach the surface, they turn into visible light long wavelength they become visible light, but these are basically the gamma rays produced in the core. And photos above photosphere is chromosphere. You can imagine 2000 to 2500 is the thickness here density is low temperature is high so it is almost fully ionized plasma neutrals are very few. Beyond chromosphere is Corona and in Corona. This is a mystery, the temperature rises suddenly in a very thin layer of transition region between chromosomes here and Corona. So we are not discussing this here, but still there is a lot of literature on this problem. The temperature has very. Okay, I just show the figure from surface, we go above temperature first decreases, then again increases remains somewhat constant and then it takes a jump. And this is the transition region through which it abruptly increases. And these are the figures about son. These are the speakers now I will apply this two dimensional theoretical model to create the speakers. This is grass like structure as grass on the earth they say so son has the grass of hot plasma. These are plasma tubes cylindrical tubes emerging all the time. And maybe I have shown right here that at any time we if we look at the surface of the sun. There are half a million speakers on this surface. So this is a very common phenomena and one needs to understand what is happening in the plasma here. So this is the two dimensional version of our exact solution, which was published in 2010. Okay, three dimensional is recent, but we don't have any application for that so far. If psi has this structure e raised to power Iota. No, mu x plus new by and plus e raised to power mu x minus new by this is the structure of psi and TJ varies linearly along Z and y direction, then we produce magnetic field. You see here in equation 50 in three dimensional. And the flow also becomes three dimensional, because we are not interested in three dimensional I just want to see the vertical flow how it is produced. So if we suppress this why variation and just consider. Okay, here is. Okay, I will discuss it later. First we take the variation of temperature just along vertical direction because this is the observed. So temperature gradient is given. We are not assuming it now it is given. We just what I think is that if temperature gradient in the system is given in astrophysical labor, it is not the laboratory scale. Here, if density gradients are produced, and they have appropriate form, then the force baroclinic effect force that gives acceleration to the plasma in vertical direction. So this we will just show here that how it happens. Then we see that. Okay, actually, one point is, you have not asked, I just mentioned here, you look at equation 51. The speakers actually have external magnetic field also, because the magnetic field is generated by the sun solar dynamics. And we have studied the magnetic field generation by beer man mechanism or if I'd are extended to include dynamics also. And the two dimensional model which I presented before this, it remains valid, I have checked it if there is a constant magnetic field, but it is just uni direction. If it has more directions, then the model does not work so if it is uni direction so we assume are suppose that these speakers have an external magnetic field along z axis, and the generated magnetic by baroclinic effects. So this is the total magnetic field in this structure, then model remains applicable. And generated magnetic field when we put the appropriate magnitudes of scale lens density gradient scale and temperature gradient scale and magnetic field turns out to be very small generated, but velocity you see at velocity 55 equation. It turns out to be along z axis. So this was the main interest velocity has only one component in the vertical direction, if this is given. So this was the interesting point now point is other that speak you'll have long length. And it is also observation that they move with constant velocity, that is the observed fact. So what I think is that in the bottom of the speak you'll be just consider a plasma slab for example, in x direction it is from zero to XM here just I, I have some time little by 510 minutes I have. Okay, these should not be parallel then problem becomes one dimensional. So I consider just one part, and yes, we assume that density falls like this exponential in x right place. So actually we should consider four coordinates, and then instead of cylinder, we have a scare emerging from the surface of from chromosphere to cover. And then what I assume is that if this is the speak you'll be considered this thickness of the plasma here acceleration produce and move up and one after the other. These structures are formed, and we have a form of speak you'll are cylindrical structure. So this was based on the Cartesian coordinates, and we assume that temperature gradient you see here t prime TZ two minus TZ one, this is the linear scale length, if we divide with the thickness edge. We estimate the gradient of temperature, and the acceleration produced, where is a not a notice to here I assume the temperature equal but ions and electron temperature is not equal. I am temperature is two times electron temperature. Otherwise, it's not but for rough estimate, we see the acceleration upward acceleration. It turns out to be 3.3 1024 centimeter per second, and on solar surface, the solar gravitational acceleration is 2.74 1024 centimeter per second scare, it means that this effect produces a larger acceleration in opposite direction. So my view point is that this structures move in upward direction great with greater acceleration than the solar acceleration, and when they leave this part and reaches here, the gradient disappears either temperature gradient disappears, or density gradient disappears. So they look as if they are moving with constant velocity, because then they are that is the observation. But here the point is that we assume cylindrical flow as scare flow are like this, but I did some work, but at that time I so this is the form of just psi and density. And this is the schematic of that and the flow is only along vertical direction with this. So now I tell you and I forgot to mention here. This is actually a better application in cylindrical coordinates. And we see that there are many conditions on psi velocity and other structures, but luckily, if we look at the psi as function of our theta. And TJ as TJ prime z linear, it remains linear because it is given. So del psi across del TJ turns out to be in the form of 58. This is, and this is the form of magnetic field on our theta plane. So this condition you see I mentioned in previous slides that this must satisfy del square psi, lambda square psi or lambda whatever we call this is some real parameter. And this equation satisfies if psi has this form given in 60, because the vessel functions obey the condition 61 and 62. So this was very fortunate that due to these relations of vessel functions 61 and 62, we can satisfy the Helmholtz equation this is Helmholtz equation. Del square psi equal to some constant psi. So this equation satisfy if psi has this form. So this is the J1 vessel function of order one cos theta. So this psi will also attain zero, I mean, not zero value, it is because and divided by and not if it is equilibrium density, so it becomes one. So at equilibrium, it becomes zero, and I just, okay, so this is the interesting point that again, we produce velocity equation 63 along axial direction. So this form of psi and temperature gradient in vertical direction produces velocity only along that direction. Therefore, it means that cylindrical plasma can have a force in vertical direction and moves upward. So I show you the Helicity becomes zero. This is the form of density. So you see this is in center cylinder here, the flow is broken. So it is not a very uniform type flow at different places the acceleration is produced and plasma moves in upward direction and form of size in this form. So psi has smaller amplitude because we assume this and oh sorry psi was log n normalized so n is exponential psi therefore density amplitude is larger. So I have presented this quickly. So if you comment I think this will be good. This will be good. Actually, this model can produce three dimensional two dimensional magnetic field, it depends upon the choice of side, but at the end, the application of the model, I could find at present. I am discussing with astrophysics, people of astrophysics, and I am thinking that maybe the chronal mass ejection can also be excellent but they say that chronal mass ejection has associated magnetic fields, large magnetic fields and magnetic reconnection etc. So this mechanism can be just an additional effect on it, but that we will look later. But unidirectional flow that is a special case, special solution. X and Y. Yes. The B naught, external magnetic field, we assume is in the Z along Z axis and it is, it turns out that its magnitude, for example on the sun is about 100 Gauss in suppose here, the generated magnetic field is very small 10 to minus seven Gauss. If you put these magnitudes. So this means that basically that is why people say that it is a seed magnetic field, magnetic field produced by Baroque-Linic effect is very small, but flows are produced are very large. So I think it is the mechanism which explains flows more dominantly than the magnetic field. Okay. Virtual particles. Virtual particles. There are three conditions of plasma and as we see in the experiment I have seen the plasma flows for a while. The plasma does not follow? Close for a while. I think Professor Mahajan maybe can explain. But what we are applying this is the, I mean plasma given by nature to us. Maybe experimentalists can answer better than us experimentalists. I have a new idea I don't understand the question. Right. I don't know what is the animation time. What is the animation time? What is the animation time? What is the animation time? What is the animation time? And in fact the animation time is the same. And what is the animation time? So I think here I am mixing the things all together. And sometimes you create the motion. And then you create the motion by making a state. These are some things that are stationary in the system. And you can give them a fun time at them. And I think they are phonons. But I mean they have got these as particle and antisposs hurdle. There are no anti-Polans. You don't need to animate their pola. At least one is best for my knowledge. So one must not extend the concepts to some of the sustained limits once you get it. And, you know, so I don't mean if you're kind of discouraged people when you're taking up new processes, you're sure that's all.