Causality and heat transport in low magnetic shear stellarators

Resumen

Humans have become more and more dependent of energy. Humankind is living, for the last century, in an era of an unprecedented economical expansion and energy has become an issue. The problem is not just find an efficient source of energy but take care of its sustainability and environmental consequences. In the last decades, renewable energies (as wind, solar panels or hydroelectric) have shown to be efficient, economically viable and environmentally friendly. This has allowed a growth in its adoption in the electricity production and many people demand a larger implementation of this kind of source of energy. However, what is not obvious for the general public is that a 100% implementation of renewable energy is not viable. Being an intermittent source of energy makes impossible to satisfy the energy demand of a country in a mid-term. Nuclear Fusion is becoming the most promising way to achieve a reliable and sustainable source of energy in the coming future with a low environmental impact. From all the possible mechanisms to reach Nuclear Fusion as a source of energy the Magnetic Confinement is the most promising one. For fusion reactions large values of temperature (hundreds of millions degrees) need to be achieved so the fuel becomes plasma. Due to the large temperatures, plasma must be confined and kept away from the walls of the reactor. Since plasma is made of ionized particles it reacts to an electromagnetic field. Under such large temperatures plasma is far away from equilibrium and high temperature gradients (and heat fluxes) appear. For several decades, heat transport has been one of the main issues to achieve the goal of an efficient nuclear fusion reactor. High losses yield to a poor confinement and reduce the viability of a reactor. Turbulence has been observed as the main element in transport in most of fusion plasmas, experimental values show confinement times up to two order of magnitude lower than Neoclassical predictions. This transport is ``faster'' and cannot be explained by diffusive (classical) theories. Different theories have been proposed for several years to try to understand turbulent transport in fusion plasmas. Some of them were more traditional (like in physics of fluids) as the calculation of effective diffusivities which then are introduced in advective-diffusion equations. Others were novel approaches as the Self-Organized Criticality (SOC) paradigm. Even though, heat transport is not yet well understood and still much work is necessary. One approach to reduce transport in fusion devices are the transport barriers. These barriers reduces considerably the turbulent transport. The physical mechanism of a transport barrier consists in a spatial region where the plasma velocity changes radially. The plasma rotation is usually caused by the presence of a radial electric field at the edge which generates an ExB flow in the poloidal direction. One of the most successful fusion devices is the stellarator. It consists in a toroidal geometry chamber where the magnetic fields are generated by external coils. Its main advantage is that it can operate in steady state and there are no currents in the plasma so no current instabilities should appear neither the possibility of disruptions. In this work we study heat transport in TJ-II and W7-X stellarators. The TJ-II is located in CIEMAT in Madrid (Spain) which started operation in 1997. The W7-X, the largest stellarator in the world, was built in Greifswald (Germany) and first plasma was launched in 2015. Even though the better design and optimization of W7-X, both devices are a stellarator with low magnetic shear which is going to facilitate our analysis. A low magnetic shear offers a low density of low order rational surfaces. Hence, the analysis of the impact of these surfaces (and the flow associated to them) in heat transport is clearer. In this thesis heat transport in fusion plasmas is analyzed in the framework of resistive MHD simulations. Chapter 2 introduces the MHD model used to simulate fusion plasmas in the TJ-II and W7-X stellarators. It is not the aim of the model to fully describe plasmas in real stellarators but to provide a (relatively easy) model where the basic plasma physics is present. The model has to be seen as a tool to simulate the fundamental physics in the mentioned stellarators and extract qualitatively some of their main properties. In the model the main instabilities are the pressure driven modes, in particular, the resistive interchange modes. Furthermore, the model includes diamagnetic effects. Reduced MHD equations are used here in the Greene and Johnson formalism, and a straight helical system is assumed within a cylindrical geometry. The model consists in four equations which are the magnetic flux equation, momentum balance equation, density equation and electron temperature equation. The model is numerically solved using the code FAR. Numerical data is analyzed using the Transfer Entropy (TE) technique which is introduced in Chapter 3. The technique is relatively new in Plasma Physics for Nuclear Fusion so it is introduced from the Information theory field and then some examples are provided in order to understand its applicability. The traditional methods (linear correlation, conditional averaging...) are conceived to identify a relation between variables. They analyze previous events and determine if they are relevant in future events. However not always the causality relation is identified or it may lead to confusing conclusions because these techniques frequently rely in our first assumption of cause-effect. Furthermore, as it is well known, correlation does not imply causation. The TE is a technique which measures the information in each signal and quantify the information propagation between two time signals. The TE can identify if previous events in one signal can be used to predict future events in another signal. Its main advantage is that it shows the direction of that information flow. In this sense, by "causality" we mean that if the information is flowing from A to B then A "causes" B. Therefore, we make an analogy between the information propagation and heat propagation. The most used methods to study heat transport in experimental fusion plasmas are the perturbative methods. They have been used during decades in most fusion devices. Most of the experiments use external heating modulation to determine the plasma response and, in this way, analyze heat transport. In numerical models is relatively easy to introduce a heat perturbation in the plasma and study its time evolution. That is not the case in experimental plasmas where to set a single pulse can be not feasible and it is not easy to identify it due to the background noise. The approach presented here, based on the Transfer Entropy, illustrates a new tool to analyze heat transport. It is applied to numerical simulations of TJ-II and W7-X plasmas. In recent experiments in TJ-II, the ERCH was used to heat inner locations of the plasma and then observe the spontaneous perturbations generated by the heating. The electron temperature was measured using ECE detectors at different radial locations during the presence of spontaneous perturbations in the core. Then the Transfer Entropy was applied from a reference point in the core to the different signals radially distributed. In this way, radial heat propagation was observed using TE. However this propagation was neither continuous nor diffusive and showed "trapping" regions where perturbations were slowed down. Chapter 4 sums up some of the experimental observations and then apply the same Transfer Entropy approach to numerical simulations of TJ-II plasmas. The numerical model allows us to understand the underlying physics and interpret the experimental results. In the simulations, a single heat pulse is set in the plasma, then its time evolution is analyzed. We can identify as well regions were the perturbations are mostly trapped and other regions were radial transport is faster. The simulations are compared with a diffusive model and, as we expected, the numerical results differs from that diffusive model. One single perturbation only allows to analyze transport in the surrounding areas because it has to be weak enough to not change the plasma in excess. Therefore, we set pulses at different locations in independent simulations. The TE results from the temporal evolution of the perturbations indicate that different radial regions exhibit different transport. The trapping regions are suggesting the presence of (mini) transport barriers. These barriers seem to be generated by the radial shear in the average poloidal flow. The same analysis is made for a different magnetic configuration. Even though the temperature, density and poloidal velocity profiles are different, the perturbative analysis using the TE shows similar conclusions about heat transport. Transport barriers are present in the different plasmas although their radial location may vary. The same study of heat transport was done in the W7-X. In spite of the fact that the W7-X device has a better performance and many differences with the TJ-II, it has a common characteristic of being a stellarator with low magnetic shear. Due to the low shear, low order modes may extend in a wider radial region. The presence of these rational surfaces may generate transport barriers. Chapter 5 introduces some of the experimental observations in the first campaign in W7-X and then our numerical model is used in order to interpret these experiment results. In the numerical simulations different heat perturbations were set at several radial locations to study transport. Using the TE approach we observe, again, radial regions where heat perturbations are trapped or regions with enhanced transport. The radial shear in the poloidal flow seems to be the reason to observe the trapping regions. In Chapters 4 and 5 the heat transport is studied qualitatively. The Transfer Entropy allows to identify radial regions in the plasma with different heat transport. However, in Chapter 6, we use the Transfer Entropy to quantitatively study that radial heat transport. The technique is able to estimate an effective diffusivity at different radial locations. This approach "averages'" transport in a certain region and estimates an effective coefficient (diffusivitty) which reproduces a similar behavior within the diffusive theory. We assume that the heat diffusivity is a local quantity and depends on the radial position. So for a small radial interval the diffusivity is a constant. These diffusivity values are obtained with the perturbative method using the Transfer Entropy. First, the approach is applied to a known diffusive model to validate its effectiveness, then it is applied to the numerical simulations of TJ-II plasmas for two different magnetic configurations. The effective diffusivities yielded from this approach are compared with the ones obtained from local quantities (flux and gradient). The results show that the Transfer Entropy is a capable technique to estimate and evaluate heat transport in magnetic fusion devices. Chapter 7 describes the emergence of transport barriers and turbulent vortices. The safety factor q illustrates the magnetic configuration of a fusion device, it is the ratio of toroidal turns by poloidal turns of the magnetic field. The q is not constant and, in general, it is a function of the radius. There are regions in the plasma where the safety factor is a rational number and in these surfaces the magnetic field lines are closed. These regions are known as rational surfaces. The model used in this work has as the dominant instabilities, the resistive interchange modes. These instabilities are nearly constant along a magnetic field line and its short perpendicular wavelength makes them to be localized in radius. Furthermore they are localized in the main rational surfaces. Due to the nature of the interchange modes, these instabilities lead to the formation of turbulent vortices with the same structure as the magnetic field lines. Therefore, those turbulent vortices may have a filamentary structure. Furthermore, these vortices and instabilities in the plasma contribute to an average poloidal flow. The poloidal flow varies along the radial direction and the radial shear of that flow can lead to the formation of transport barriers. Then, we find that where we observed trapping radial regions (Chapter 4 and 5), there are filamentary structures. We associate the presence of these filaments with the transport barriers. The turbulent vortices, related to the transport barriers, should be found as filamentary structures in the plasma. In Chapter 8, the topology of filamentary structures in TJ-II is analyzed using the Transfer Entropy technique. First, some of the experimental results in TJ-II are summarized and then our numerical simulations are used to interpret these results. Focusing on the electrostatic potential we can observe that there are (filamentary) structures following the magnetic field lines in radial surfaces. Filaments are very long structures which extend several meters along the fusion device. Note that in a poloidal cross section filaments seem to be separated a few centimeters. However, the connexion length between two points along the filament may be several meters long. Usually, in experiments, two (or more) detectors are placed at different spatial locations. The experimental detection of these filaments can be a bit complex because the low number of measuring points. On the other hand, numerical simulations allow to explore all positions. The Transfer Entropy is applied to the numerical results at different locations in the plasma. As the filaments are rotating they are going to cross first the reference measuring point and then the other measuring points (or the opposite depending on the poloidal velocity direction). This effect can be detected using the transfer entropy. The technique is able to detect the periodicity of the filaments and at the same time their poloidal velocity (and direction) can be obtained. The filaments, as we expected, are rotating with the same velocity as the plasma. Furthermore, analyzing the numerical data in different radial locations, we are able, using the TE, to estimate a radial width of the filamentary structures. The transfer entropy method can detect the connexion path between two probes only if the filament structure is not broken in more than two pieces. If so, there may not exist a path to connect both probes and the TE would not work. In this work filaments appear not to be broken. Finally, Chapter 9 reproduces some of the studies from the previous chapter but for the W7-X device. Using our numerical simulations, some of the properties of the filamentary structures are studied and similar conclusions are obtained. Very long filamentary structures are found in our W7-X plasmas. Obviously, their distribution and location is different to the previous results in Chapter 8. We focus on measurements of the electrostatic potential at different spatial points. The TE is able to detect when a filaments cross different measuring points. We obtain, using the Transfer Entropy, the periodicity and the poloidal velocity of filamentary structures. Their poloidal velocity coincides with the plasma poloidal flow. The radial extension of the filaments is also estimated using the TE. At this moment, there are no studies about filaments in the W7-X. However, this work may be used in future experiments to interpret the results. In this thesis, the Transfer Entropy has been presented as a new technique to analyze heat transport in magnetically confined fusion devices. It has been shown to be a valuable tool to identify and quantify heat transport. Furthermore, it has been used to detect filamentary structures in the plasma and to obtain some of their properties as periodicity, poloidal velocity o radial width.