Collisionless magnetic reconnection : the Contour Dynamics approach

A long time ago, mankind was a pretty pathetic species. We were cold, dependent on the sun for light and heat, and our diet was abhorrent. Fortunately, Prometheus felt sorry for us, and presented us the gift of fire. Zeus was not too keen on letting humans set fire to just about everything they could, and Prometheus paid dearly for his act of compassion by being chained to a mountain ridge in the Caucasus, where his liver got eaten out every single day by a large eagle, after which it grew back on again. Mankind, on the other hand, were having a ball [1]. To a certain extent, this is still the situation today. Mankind sets fire to anything that provides comfort, to make it warm when it is cold, to make it cold when it is warm, using small explosions inside cylinders of a car to make a baby sleep because it likes the humming sound. And of course, Prometheus would still be up there somewhere if it were not for Heracles. But in this day and age, we start to feel that the gift of Prometheus may no longer be enough to lavish our lust for energy and we start to realize that we need to redirect our looting frenzy towards a new source of energy if we want to sustain the luxury and abundance as we have known the last decades. Of some sources we can see the bottom, of some we are not sure under what restrictions we will be able to buy them, and in general we can see the side-effects of the waste that we produce. Therefore, we turn to Helios, the Sun. Not to become dependent once again on when and where he will shine, but to usher him to reveal his secrets, so that we can make a sun of our own. He has been relatively forthcoming in providing us with the theory of what is supposed to happen: in a process called nuclear fusion two light atomic nuclei collide and merge into a heavier one. The reaction products carry the energy that is released in such a process as kinetic energy, i.e. by going extremely fast. The process that seems the most likely or feasible candidate to produce fusion power in a reactor here on earth is between two hydrogen isotopes, deuterium (D) and tritium (T) to yield helium (what’s in a name) and a neutron,21D + 31T ¿ 42He + n + 17.6MeV. Deuterium is a hydrogen isotope, and consists of a nucleus with a positively charged proton and a neutron and a negatively charged electron surrounding it, whereas tritium has a proton and two neutrons in the nucleus, which makes it unstable, so that it suffers from radio-active decay. If we know what the reaction is, what is stopping us? The resources will not run out for the next couple of thousand years, as lithium is widely available to breed tritium from the neutron shower inside the reactor, and deuterium can be extracted from sea water easily for many more thousands of years. Also the reaction products are relatively clean: helium is used to fill balloons at children’s parties, and the neutrons do cause radio-active activation of the steel of which the reactor is built, but this is mildly radio-active and can be safely stored. By blowing up innocent islands in the southern Pacific in the early 1950s it has been proven in a most dramatic way that nuclear fusion does indeed work, and ever since the challenge has become to master this process so that we can make it evolve in a controlled and orderly, non-violent fashion. This is non-trivial, because the nuclei need to overcome the electric repulsion, or the Coulomb interaction, to get close enough to merge. This requires a very high mean velocity, or equivalently, temperature. At these high temperatures the 135 electrons are separated from their nuclei, and the fuel mixture becomes a (hot) plasma, in which the nuclei, or ions, and electrons can move independently but under the influence of each others electromagnetic field. This brings us back to Helios once more. Confiding in us as far as the energy source was concerned may have seemed relatively safe. Zeus’ wrath is not yet upon him, as for us it is not a matter of fusing the particles, but how we do this to create an efficient cycle. The Sun confines plasma in its core at roughly 15 million degrees Kelvin, with its huge gravity field pulling the plasma inward, which leads to a relatively slow, smouldering reaction. We have to do better. And hotter. We want to maintain a fusion reaction with minimum input power and maximum yield, which means that we have to heat the D-T plasma to 150 million degrees Kelvin. To keep it there, away from cold, melting walls, our present day fusion reactor concepts make use of a magnetic field. Charged plasma particles in a magnetic field describe gyrating orbits around magnetic field lines, so if the magnetic field does not touch the walls of the containing vessel, neither will the particles. This can be achieved by giving the vessel and the magnetic field a toroidal shape. The best candidates for delivering net fusion power in the near future is called the tokamak, a machine that is based on producing a strong, dominant, magnetic field in the toroidal direction (the long way around the torus), and the ability to draw a current through the plasma, either inductively or non-inductively, resulting in a smaller poloidal component. The now helical magnetic field lines are thus neatly organized on nested concentric surfaces, also called flux surfaces. Here is where magnetic reconnection enters the story. When the magnetic field would conserve this nested topology, like a toroidal magnetic Russian matruschka doll, the confinement of the particles and the heat would be excellent. But it does not. With all those particles racing along magnetic field lines, the plasma actually becomes a wildly turbulent, swirling mass, pushing those flux surfaces to and fro. When magnetic field lines that are not aligned are pushed together, not only pressure builds up, but the magnetic tension of the field as well, resulting in a large electric field hurling the electrons in place to shield the surfaces from one another. If somehow this sheet of electrons does not do its shielding job properly, the magnetic field finds its own way to release the pressure, by untying the magnetic field lines and connect them to other field lines in a way that there is less magnetic stress. This is called magnetic reconnection. It happens when e.g. the current sheet between the flux surfaces degrades by resistivity, so when the electrons and the ions collide so much that the sheet becomes too weak, or by turbulent eddies dislocating the meandering electrons. The once neatly organized flux surfaces break up, and what used to be in- and outside the flux surface is no longer clear: regions appear that no longer belong to the original surfaces, and the new boundaries are now called separatrices. In a toroidal plasma these regions form helical ribbons around the original surface, and when one looks at a cross section of the plasma, they can be identified as a chain of magnetic islands, with an o-point in the middle, and an x-point where the field lines break open and reconnect again. The release of magnetic tension then may result in a pressure drop, sucking more plasma and magnetic field into this pit of magnetic annihilation, and thus the process can feed itself, in a vicious cycle that because of the tearing of the flux surfaces is called a tearing mode. The effects of such a process are destructive for the confinement of hot particles that should stay in the core of the plasma to eventually fuse together, which is the whole idea behind a fusion reactor. It may result in (more) turbulence, it may induce other flux surfaces to start tearing up, and when the magnet islands start to overlap this leads to magnetic field ergodization, which means that field lines become randomly displaced from their original flux surface. They can fill up a three dimensional volume, and because particles race along them, they can race out of the plasma (not the other way around, because the pressure inside is much higher). When the islands grow really large, chances are that the plasma ends in a disruption. All the energy that was stored in the magnetic field is released at once, leading to considerable damage to the fusion reactor. This may sound as if we have a pretty good idea of what is going on during reconnection inside a tokamak plasma. But, as was mentioned before, a fusion plasma is very hot. Very hot. In very hot plasmas the electron mass plays an important role in causing of magnetic reconnection. In that case, we call this process collisionless magnetic reconnection, as in cooler plasmas a more straightforward reconnection mechanism is more important, caused by the collisions between the plasma particles. It was shown that collisionless reconnection can explain the fast reconnection rates that have been measured in the centre of tokamak plasmas. The model that is studied in this thesis relies on this mechanism for magnetic reconnection. To model the effect that electrons do not collide but can have different temperatures, the equations that determine the dynamics of electrons in a plasma with a dominant magnetic field in one direction, make use of a distribution function that specifies how many electrons have a certain velocity parallel to the magnetic field with respect to the bulk of the plasma. They look like the equations that determine vorticity (the amount of rotation) in a two dimensional, shallow, fluid system. Vorticity is generally dragged along or advected by a stream function, or a flow field, like storms are advected by the wind. The difference is that in this equation, the drift-kinetic equation, for each parallel velocity we have a different type of vorticity, advected by a different stream function. To keep the metaphor of the weather, it is analogous to having a different wind for each height, where height would correspond to parallel velocity. This correspondence is very real: indeed storms are advected differently at different heights, and are described by very similar equations. This observation opens the way to make use of similar tools as in fluid theory to study plasma, such as contour dynamics. Vorticity in two dimensions can be thought of of being built up like a topographical map: a mountain is described with a line or contour that corresponds to 100 meter altitude, and within this contour lies another contour corresponding to 200 meter altitude and so forth. The same is true for vorticity. One can draw a contour that corresponds to a certain amount of vorticity, and within a contour that corresponds to more vorticity etc. The locations of these contours may change in time, as vorticity evolves, but it is clear that these contour lines cannot cross: on a map one can think of strangely shaped mountains, but with vorticity this is just impossible. For plasma, we can draw these contours for every different type of vorticity, i.e. for every parallel velocity. For different types, they are allowed to cross, for the same type they may not. Very important is the fact that in this way, all of the plasma dynamics is now captured by the location of the contours, and the interaction between them and themselves. Back to parallel electron velocities. A contour now describes the jump in the parallel electron velocity. If we construct an area where electrons move with respect to the bulk in a certain direction, this is equivalent to saying that there is a current density. Actually, it is a bit more subtle, because electrons interact with the plasma by their electrical charge and, if they move, by the magnetic field they generate, so we have ‘pure’ current if n less electrons move in the negative direction and n electrons more move in the positive direction (or vice versa). If we construct a straight or annular layer with a current density in this way, this layer Figure S.1: From left to right: the contours of the electrons with the highest velocities for a simulation where the mode number of the perturbation is m = 5, and the distribution function is replaced by N = 5 contours that mark the boundaries of the 5 corresponding parallel electron velocities. The middle figure shows the isolines of the electrostatic potential (or the altitude lines of the electric field) along which slow electrons drift, and the right figure shows the isolines of the magnetic potential, or a top view of the flux surfaces along which the magnetic field is confined. Fast electrons tend to move along these lines. stays just as it is. Nothing happens. All the equations need an excursion from symmetry, because there is no gradient or gradual change anywhere, except at the location of the jumps. We can only excite or perturb the system at these contours, and then calculate what happens. There are three possibilities: when the contours are dislocated they can bend back again, resulting in a (damped) wave, like in a guitar string. Then the system is called stable. The contours can be moved and still nothing happens, like a ball on a horizontal plane, which is called metastable. But if the contours are moved and they start moving in the same direction of the perturbation, the perturbation grows exponentially. The system is then linearly unstable. In Chapter 3 and 4 the stability of an equilibrium that consists of a straight slab of plasma with in the middle a region with an electrical current is investigated. It is found that it can be unstable with respect to a tearing mode, and only to a tearing mode. This leads to a chain of magnetic islands through the middle of the current layer, whose growth rate depends on the amount of current, the magnitude of the magnetic field and the electron mass and density. The equation that links the linear growth rate to the other parameters is the linear dispersion relation. When a temperature difference across this layer is applied, having warmer plasma on one side of the current layer than on the other, the island chain starts to move in the direction perpendicular to both the magnetic field and the direction of the temperature gradient. And if the islands have already obtained a finite width, and the dynamics of the island becomes dependent on the width of the island, so in the nonlinear regime, a shift with respect to the external perturbation remains. This shift is important, because if there are multiple chains of magnetic islands next to each other, the local temperature gradient will depend on the location of the islands. This can lead to self-organization of magnetic islands, so that they fit better, or so that they start to overlap sooner. The onset of magnetic field ergodization can depend on the way these islands shift with respect to each other. The magnetic island becomes ‘droplet’ (or onion) shaped, because the x-point is shifted more than the o-point. In Chapter 5, the linear stability analysis of Chapter 3 and 4 is applied to an equilibrium that is circular instead of straight. This was done for the two-fluid variant of the drift-kinetic equation, using only two contours to describe the perturbed parallel velocity of the electrons. For this shape of equilibria, the computer code that was developed in Eindhoven could be used to calculate the evolution of those contours in time. The growth rates that we found in this way showed good agreement with the theoretical predictions made by the linear dispersion relation. The numerical code is fully nonlinear, which means that it can calculate the evolution of the contours self-consistently, without assuming that they only move a little bit. One purely nonlinear effect is that after a certain amount of time, the reconnection stops, or in other words, the magnetic island saturates. And though the two-fluid model is essentially isothermal, it is possible to study the effect of the ratio of the thermal velocity of the electrons and the mode velocity with which the island grows. The simulations show that when the electrons move fast compared to the velocity of the mode, they move more along the separatrices of the magnetic island, following the magnetic field lines, whereas if they are slow compared to the mode, they drift along isolines of the electrostatic potential, perpendicular to the electric field. The numerical code was extended so that we could use more than two types of contours, and simulate the effect of a temperature gradient on the process of collisionless magnetic reconnection. The rotation of the island chain with the diamagnetic velocity and the deformation of the magnetic island are found to be in pretty good agreement with the predictions that were made in Chapter 3 and 4. The x-point, that rotates faster than the o-point because there the same temperature difference occurs over a shorter distance, is observed to decelerate again after some time. This is attributed to the appearance of a region with negative current density that pushes the x-point away from the o-point, because the o-point corresponds to a region with positive current. This concludes the work that was done for this thesis, which was done as part of a project that should contribute to a working nuclear fusion device. But browsing through the contents of this thesis the image of a power plant may not surface very often. However, magnetic reconnection can lead to some of the most violent phenomena in magnetized plasmas, that need to be controlled if we want to use nuclear fusion as an energy source. It has the power to reshape the magnetic configuration within the plasma, ultimately destroying confinement altogether, and it may have the subtle, even useful effect of removing the Helium ‘ash’ out of the core of the plasma, saving it from extinction. It is essential to have a deeper knowledge of this process to be able to control it or prevent it from happening. We have to understand how it works, what the electrons that are involved in this process do and how they influence the evolution of the reconnection process, and how important parameters like the temperature gradient may play a role, keeping in mind that a tokamak generates the largest temperature gradient known in the universe. This was the main question that was addressed in this thesis.