Background and state of the art

The confinement of hydrogen isotopes by means of toroidal magnetic fields with nested magnetic surfaces represents one of the routes towards electricity generation based on controlled fusion energy. In magnetic confinement fusion reactors, the fuel (a mixture of deuterium and tritium for the first envisaged reactors) is injected in the confinement region and heated. Then, the atoms are ionized, and the nuclei and the electrons (being particles electrically charged) interact with the confining magnetic field. This globally neutral gas of electrically charged particles is called plasma. When the temperature of the plasma is high enough, fusion reactions start to take place. According to the Lawson criterion, these exothermic reactions will be sufficient to overcome the energy losses and maintain the temperature of the plasma if the so-called triple product (the product of electron density, ion temperature and energy confinement time) exceeds a certain value.

The two most developed concepts for magnetic confinement fusion reactors are tokamaks and stellarators. The main difference between them is that the tokamak magnetic configuration is axisymmetric, whereas the stellarator magnetic field is intrinsically three- dimensional. The tokamak line is more mature and, actually, the first experimental fusion reactor, ITER, will be a tokamak. However, the stellarator is a promising alternative for next- generation devices because it does not suffer from some of the problems that tokamaks have. These problems have their origin in the fact that an important fraction of the tokamak axisymmetric magnetic field is generated by inducing a large toroidal electric current in the plasma. This is problematic because it can drive magnetohydrodynamic instabilities and because the need to generate this current by induction makes steady-state operation difficult. Stellarator magnetic fields, on the contrary, exploit their three-dimensional nature to generate the magnetic field by means of external coils only. The challenge for stellarators is to identify and design three-dimensional configurations that have transport levels (and, consequently, energy losses) as low as axisymmetric configurations have. This project addresses several problems in the area of transport in stellarator plasmas that are of the utmost importance for the development of this confinement concept. The theoretical tools, numerical codes and experiments involved in the proposal are at the cutting edge of the efforts of the international stellarator community.

Two types of transport processes lead to substantial energy losses in stellarators. First, the combination of inhomogeneities in the magnetic geometry and binary particle collisions gives rise to the so-called “neoclassical transport”. Second, the (turbulent) fluctuations triggered by the self-consistent coupling of the plasma and the electromagnetic fields are responsible for what is known as “turbulent transport”.

In stellarators, neoclassical transport is dominant (whereas in tokamaks it is quite small for high temperature plasmas) in many fusion-relevant situations [Dinklage-13] and, for this reason, the design of stellarator magnetic fields has been focused so far on the overall minimization of neoclassical energy transport of the ions, in the hope that this will lead to maximization of the energy confinement time once the device is built and operated. In the triple-product, the density is typically considered a parameter that can be adjusted by means of fuelling and the ion temperature, for a given input power and density, is positively correlated with the energy confinement time.

However, bringing the stellarator configuration to maturity as a possible long-term alternative to tokamaks is one of the missions of the European Fusion programme [Eurofusion-12], and it is now clear that achieving this goal requires a more comprehensive approach than just minimizing ion energy neoclassical transport. Not only should the optimization and design strategies be validated experimentally and, if necessary, improved, but they should ultimately lead to reactors that can actually operate in steady-state conditions [Eurofusion-12]. This approach necessarily implies dealing with the multi-species (“a species” refers to each type of particle present in the plasma, characterized by its electric charge and mass) nature of fusion plasmas, and also with a simultaneous understanding of both branches of transport: neoclassical and turbulent. Let us explain why.

When studying the energy balance of a fusion device, electron dynamics can often be neglected, being possible to consider only ion energy transport, due to the small ratio of electron to ion mass. However, in stellarator scenario design, the density cannot be used as a fixed parameter as in e.g. [Turkin-11,Geiger-15], and particle density balance has to be carefully studied as well. The reason is that the hydrogen isotopes are supplied typically from the edge (although in some scenarios they could be injected at inner positions by means of pellets) and this fuel needs to make its way up to the central region, where the temperature is high enough, in order for the fusion reactions to actually take place. This makes electron transport an important piece of stellarator modelling: on spatial scales larger than the Debye length, plasmas are neutral and, in a plasma composed of electrons and hydrogen isotopes, with only trace amounts of other species, the electron density must equal the ion density at all times; for given density and temperature profiles, this means that a radial electric field will arise in the plasma volume in order to make the ion and electron fluxes equal. When the ion and electron temperatures are similar, due to the larger mass of the ions, this radial electric field tends to be negative (i.e., directed towards the core), in a scenario called “ion root”, inwhich it is sometimes said that the ion particle flux level has been brought down to that of theelectrons, which are then called “rate-controlling” species.

Another mechanism that may lead to important energy losses and thus might preclude the achievement of steady state operation of a fusion reactor is radiation by impurities, i.e., by atomic species other than the fusion reactants (in general, when we use the word “ions” we refer to the nuclei of the reactants and when we use “impurities” we refer to positively charged states of the impurity species). The presence of even small concentrations of impurities (especially those having high charge number Z) in the confinement volume may have deleterious consequences on plasma performance. Impurity sources are mainly located at the stellarator edge, since high-Z species ranging from carbon to tungsten are used as divertor and first wall materials, thus far from the region of highest temperature. Nevertheless, inward transport of the impurities, usually caused by the negative radial electric field, may accumulate them at the core, where they absorb energy and radiate it, causing fuel dilution and power loss.

Consistent treatment of the different particle species present in the plasma is therefore fundamental for the characterization and design of transport in fusion-relevant scenarios. In a plasma with a trace amount of impurities, we have seen that equality of the ion and electron fluxes determines the radial electric field. This radial electric field is in turn one of the so-called “thermodynamic forces” driving not only impurity transport, but also ion and electronenergy transport. Furthermore, equality of the electron and ion density at every single point of each magnetic surface leads to the formation of electric fields that are tangent to these surfaces, and that affect as well the radial transport of all species, specially impurities.Finally, when the amount of impurities is large enough (i.e., they are “non-trace”), they have an effect on the transport of main species (e.g. hydrogen ions and electrons), directly by making them more collisional, and also indirectly by contributing to determine the electric field on the plasma volume.

It is necessary to understand and compute the two branches of transport mentioned above, neoclassical and turbulent, in order to accurately calculate particle and energy transport of the different species in the whole plasma. Due to the unfavourable temperature scaling of the neoclassical fluxes, energy and particle transport are typically determined by neoclassical processes in the core of stellarators [Dinklage-13, Velasco-16]. As a consequence of the presence of a negative radial electric field, collisional processes usually determine impurity transport as well [Burhenn-09]. However, validation of neoclassical modelling is far from perfect, and quantitative disagreement between predictions and experiments is still found on many occasions [Dinklage-13, Satake-14, Velasco-14, Tallents-14, Garcia-Regaña-18].

Some of the publications referenced in the previous paragraph have been part of two previous projects, ENE2012-30832 and ENE2015-70142-P, whose research team consisted of essentially the same members that participate in this proposal. The latter was strongly focused on the validation of standard neoclassical tools and on the derivation of neoclassical theory aiming at the characterization of transport in the specific case of optimized stellarators. Additional work on these topics carried out under ENE2015-70142-P appears in references [Calvo-17, Calvo-18, Velasco-18]. A relevant part of the present proposal of this project deals with the derivation of further extensions to the theory and to the development and validation of a fast simulation code (named KNOSOS) that makes use of these extensions. The multispecies approach and the interconnection between neoclassical and turbulent transport are the main focus of this proposal. The goal is to have a fast and accurate numerical tool that brings the neoclassical predictions closer to the experiment. Once validated, such a fast code can be used for improved stellarator optimization.

There are situations, however, in which neoclassical predictions fail completely to provide accurate results, and it is not expected that improvements in the theory will make the situation better. For instance, in the edge region of stellarators, the temperature drops, and the neoclassical transport predicted for the main species is well below the total transport levels experimentally measured. Even in the core of certain stellarator plasmas, neoclassical theory predicts a depletion of particles that is not actually observed in the experiment [Maassberg-99]. Finally, with respect to impurity transport, two promising (but poorly- understood) scenarios have been found in which low impurity content coexists with an inwards-directed radial electric field: the HDH mode of the stellarator Wendelstein 7-AS [McKormick-02] and the impurity hole of the Large Helical Device [Ida-09] respectively. More generally, even in some situations in which impurity accumulation or expulsion is correctly predicted by neoclassical theory, the impurity diffusion (i.e., the flux proportional to the impurity density gradient) is overestimated by up to an order of magnitude [Langenberg-17, Huang-17]. All these situations, that cover energy, particle and impurity balance, three essential topics for the characterization of stellarator scenarios, probably need accounting for turbulence to be accurately described.

Gyrokinetics [Catto-78] is the reduced kinetic theory that describes microturbulence in plasmas. In the last years, under the projects ENE2012-30832 and ENE2015-70142-P, we have carried out a characterization of electrostatic instabilities in the stellarator TJ-II by means of gyrokinetic simulations [Sánchez-14], including first comparisons with experimental measurements [Sánchez-17, Sánchez-18]. In the present project, we propose to use the gyrokinetic codes estella [Barnes-18] and EUTERPE [Jost-01] to simulate instabilities and the process of turbulent saturation in stellarators. Gyrokinetic theory and simulations in stellarators are less developed and understood than their neoclassical counterparts, and the general goal of this part of the project is to assess what extensions of the theory and simulation can be of particular relevance for obtaining accurate gyrokinetic simulations in stellarator geometry, and contribute with gyrokinetic simulations and theoretical developments to fill those gaps.

In particular, let us mention that, although neoclassical and turbulent transport are routinely treated as additive, they may not be so under some circunstances. In three-dimensional systems, the usual Maxwellian approximation of the equilibrium distribution function in gyrokinetics can be violated in low collisionality regimes, for which the size of the neoclassical correction to the distribution function approaches that of the local Maxwellian. The two main lines of research of this project will ultimately lead us to first-of-its-kind multi- species kinetic (neoclassical and gyrokinetic) simulations in stellarator geometry.

These simulations will be used to study the three physics topics that have been outlined above: minimization of energy transport, density control and avoidance of impurity accumulation.


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