Theory developments

In the last recent years, we have obtained analytical results on stellarator neoclassical theory that are the basis for the goals of this task and for some of the numerical goals of task (2).

In [Calvo-17], we derived orbit averaged low collisionality neoclassical equations that rigorously retain the component of the magnetic drift tangent to magnetic surfaces, so that they capture the so called superbanana-plateau regime, and are therefore valid for very low collisionality and small value of the radial electric field. These equations also incorporate the component of the electric field tangent to magnetic surfaces and allow to determine it self-consistently. The set of equations, consisting of the drift-kinetic and quasineutrality equations, is linear and radially local, and has been implemented in the new code KNOSOS (see task (2)), that for the moment can treat model magnetic fields corresponding to sufficiently optimized stellarators. In [Calvo-18], we derived analytical scalings of the tangential electric field in different asymptotic regimes and validated them with KNOSOS for model optimized configurations.

In [Calvo-18b], we studied the impact of the tangential electric field on radial impurity transport in stellarators when the impurities are collisional and the main ions have low collisionality. We gave an analytical expression for the impurity flux when impurities are diluted and proved that even for small values of the tangential electric field, its effect on impurity transport can be large.

The objectives of this proposal under this task are:

  • Employ the techniques used in [Calvo-18b] to derive analytical expressions for the impurity flux in other potentially relevant situations. For example, when the pressure anisotropy of the main ions cannot be neglected. We will also explore the possibility to generalize these calculations to the case in which impurities are non-trace (i.e. they are not diluted in the main plasma).


Go to the bibliography.

Development and benchmarking of KNOSOS code for multi-species plasmas of arbitrary geometry

Following the recent theoretical results discussed in the theory section, we have developed the numerical code KNOSOS (KiNetic Orbit-averaging SOlver for Stellarators) [Velasco-18, Calvo-18]. KNOSOS is the first radially-local code to rigorously retain the tangential magnetic drift in the calculation of ion neoclassical transport. In [Velasco-18] we proved that this is crucial in order to calculate accurately the tangential electric field created by the bulk ions, and the corresponding radial impurity transport, especially in the case of plasmas close to impurity screening [Velasco-17], such as scenarios displaying impurity hole of the Large Helical Device [Ida-09].

However, in its current version, KNOSOS solves the set of equations with some approximations. First, it solves the equations for the simplified case of a model magnetic configuration close enough to omnigeneity. Second, it solves only the ion drift kinetic equation, and considers adiabatic electrons in the ambipolarity and quasineutrality equations. Third, it employs a pitch-angle-scattering collision operator.

The objectives of this proposal under this task are:

  • Extend and benchmark KNOSOS so that it can treat general stellarator geometries. Some first calculations of the drift kinetic equation without tangential magnetic drift and tangential electric field have been done and compared with the neoclassical code DKES [Velasco-18b]. We will include the tangential magnetic drift and we will solve consistently the ambipolar and quasineutrality equations.
  • Extend and benchmark KNOSOS so that it becomes a multispecies code. First, the electron drift kinetic equation will be solved in parallel to that of the ions. For some plasma regimes, this may require upgrading the collision operator. Second, the electrons will be included in the ambipolarity and quasineutrality equations. Depending upon other results, so will the collisional impurities in the non-trace limit.
  • Fine tune KNOSOS so that it can provide very fast accurate neoclassical estimations of transport. The set of equations solved by KNOSOS is suited to a very high degree of parallelization (in species, radial position, value of the pitch-angle variable at which the bounce-averages are calculated, value of the radial electric field, etc.). Furthermore, for each calculation, depending on the quantity of interest and the plasma parameters, a different level of sophistication may be required (e.g., the electrons may or not be considered adiabatic, or the tangential electric field may or not be neglected, etc). We will try to obtain systematic rules that speed up calculations.


Go to the bibliography.

Optimization of neoclassical transport

The viability of the stellarator programme as an alternative to tokamaks in the path towards the fusion reactor relies on our capability to design magnetic configurations whose neoclassical radial energy transport is as low as that of tokamaks. Stellarator configurations are typically designed including the minimization of the effective ripple [Nemov-99] as an optimization criterion. However, according to the experimental data existing in the International Stellarator/Heliotron Confinement DataBase (ISHCDB), the correlation between effective ripple and device performance is not satisfactory [Yamada-05]. In principle, this could appear to be at odds with the fact that neoclassical theory is able to provide reasonably good (although improvable) quantitative predictions for ion radial energy transport of stellarators for fusion-relevant plasmas, as the ones we characterized in [Dinklage-13]. One of the reasons is that the effective ripple provides the level of transport in the 1/nu neoclassical regime, but it does not carry information about other relevant regimes (and, in particular, of the transport associated with the theory refinements discussed in this project.

The objective of this activity will be:

  • Use KNOSOS to characterize more accurately the energy confinement of several devices of the ISHCDB. In particular, we will assess which of the refinements developed by the research team (and not included [Dinklage-13]) are more relevant. We will also explore the possibility of using a figure of merit for stellarator optimization that works better than the effective ripple.

Scenarios free of impurity acccumulation

The analytical expressions of [Calvo-2018] have been combined with the equations solved by KNOSOS for the main ions. This combination provides fast calculations of the radial flux of collisional impurities as a function of, among other things, the background plasma.

The objective of this activity will be:

  • Use KNOSOS to explore systematically the parameter space (including magnetic configuration, collisionality and shape of the background profiles) in order to find scenarios in which impurities do not accumulate.


Go to the bibliography.

Influence of global effects and neoclassical physics on stellarator turbulence and zonal flow dynamics

Under this task, we would like to understand what is the correct computational approach (perhaps depending on plasma parameters) to study stellarator plasma turbulence and zonal flow dynamics in multispecies plasmas.

Influence of global effects on zonal flow dynamics

The flows produced by the component of the turbulent fluctuating electric potential that is constant on each flux surface and varies across it on the gyro-radius legth scale are called zonal flows, and they are thought to play an important role in reducing turbulent transport. In [Mishchenko-08, Monreal-16, Monreal-17] the properties of long-term linear and collisionless zonal flow relaxation in stellarators were studied, deriving analytical expressions for the zonal flow residual level and the zonal flow oscillation frequency. In [Monreal-16, Monreal-17], those analytical expressions were evaluated numerically in several magnetic configurations and compared to gyrokinetic simulations with GENE (in full flux-surface domain) and EUTERPE (radially global), finding a very good agreement. A detailed comparison between calculations in radially global and flux-tube domains has not been carried out yet, but there are indications, both from theory [Monreal-16, Monreal-17] and also from simulations [Smoniewski-18], that the zonal flow properties cannot be properly accounted for in a flux-tube. Nevertheless, because of its numerical and computational advantages, this domain is often employed for simulating physics situations in which the zonal flow dynamics is considered important (see e.g. [Nakata-17], [Plunk-17]).

An intuitive explanation why flux-tube simulations might not be enough to determine either zonal flow dynamics of turbulence properties in stellarator geometry comes from noting that the radial electric field plays a key role in stellarators, with the associated ExB flow modifying the particle trajectories along the flux surface and moving the particles across field lines. This affects in particular the zonal flow linear properties [Sugama-09, Mishchenko-12] as well as the mode structure of microinstabilities [Riemann-15]. This suggests that at least a full flux surface domain is required to properly treat the electric field in a stellarator. Furthermore, the different boundary conditions imposed in each computational domain can also make a difference between radially local and global domains.

The objective of this activity is:

  • Determine the minimal computational domain required for an accurate description of the linear zonal flow dynamics and characterize the effect of an insufficient domain in the non- linear turbulent fluxes.

Influence of neoclassical transport on turbulence

As explained in [Calvo-17], one can find stellarator magnetic configurations and plasma regimes in which the deviation of the main ion neoclassical distribution from a Maxwellian distribution is as large as the Maxwellian distribution itself. In these situations, the neoclassical equations are radially non-local, radial turbulent transport is negligible compared to radial neoclassical transport and the question about the influence of neoclassical effects on turbulence is also irrelevant. We are interested in situations in which the deviation of the main ion neoclassical distribution from a Maxwellian distribution is large, but not that large. The arguments of [Calvo-17] indicate that these situations exist, for example, in large aspect ratio stellarators (all stellarators in operation today have large aspect ratio).

It is also important to note that large neoclassical corrections to the distribution function do not necessarily imply a similarly large neoclassical flux. This is for example the case of the “deep √ν regime” (i.e. the √ν regime at sufficiently low collisionality): the neoclassicalcorrection to the Maxwellian is large (in particular, its size does not depend on the collisionality) but neoclassical transport gets smaller as the collisionality is decreased, and turbulent transport could therefore gain relevance.

Our objective can be summarized as follows:

  • Characterize in what conditions, how and to what extent the neoclassical corrections to the Maxwellian distribution affect the linear mode spectrum and turbulence in stellarators. The expected outcome of this novel investigation is the identification of those conditions and the fundamental mechanisms of this interaction. Numerical tools will be developed that will be applied in other parts of this proposal and can potentially be used for the modelling of experiments at later stages.


Go to the bibliography.

Turbulent transport of impurities

Impurity accumulation is considered to be a crucial issue in the path towards a fusion-grade stellarator reactor. The reason is that stellarator operation scenarios usually display a negative radial electric field that, according to standard neoclassical theory, should cause an inwards impurity pinch roughly proportional to the charge number Z. This impurity accumulation is observed in the experiment, with very few exceptions [Burhenn-09]. One of these exceptions is the so-called “impurity hole”, observed at low collisionalities in the Large Helical Device, in which a negative radial electric field coexists with a strong outwards- directed impurity pinch [Ida-09]; other well-known exception is the High Density H-mode of Wendelstein 7-AS [McCormick-02]. With the goal of understanding these scenarios, as a necessary step prior to trying to reproduce them in a future stellarator reactor, we have implemented several extensions to neoclassical theory in numerical codes [García-Regaña- 13, García-Regaña-17, Calvo-17, García-Regaña-18, Mollén-18, Velasco-18] (see also [Helander-17]). However, none of the simulations performed so far has been able to explain e.g. the impurity hole. In parallel, as mentioned before, first gyrokinetic simulations of impurity transport in stellarators have been performed [Mikkelsen-14, Nunami-16], also unsuccessfully. We note that in these works, the simulation domain was a flux tube, and the neoclassical background was not considered; however, for example, the low-collisionality plasmas that display impurity hole are in what we have called “deep √ν regime” [Velasco-17].

The objective of this task is:

  • Determine the sophistication in the treatment of the geometry and equilibrium distribution functions that is required to accurately model the transport of light and heavy impurities in low ion-collisionality stellarator plasmas. We will perform the first turbulent simulations that include these sophistications. A systematic comparison of the so-calculated turbulent fluxes with neoclassical ones is an expected outcome of this activity.


Go to the bibliography.

Experimental validation of theoretical predictions and numerical simulations

Three running stellarators are available in the frame of the present project for planning, running and analysis of experiments: TJ-II at CIEMAT, in Spain; the Large Helical Device (LHD) at the National Institute for Fusion Science (NIFS), in Japan; and the Wendelstein 7-X (W7-X) stellarator, at the Max-Planck Institute for Plasma Physics (IPP) in Germany. W7-X is currently under the operation phase OP1.2 [Sunn-Pedersen-15], and it will later undergo an upgrade in preparation of the long-pulse operation phase OP2, which will start in early 2021. The theoretical and computational developments that are subject of this project will allow a first principle interpretative modelling of the transport measurements of the plasma main ions and impurities. In this respect, it is important to highlight that the three devices are equipped with different sets of diagnostics that provide observables particularly suitable for the validation of the simulations.

The following objective is in line with the present task:

  • Simulation of the ion heat flux (both turbulent and collisional) in the core of W7-X for several magnetic configurations and the radial particle flux of highly charged impurities with arbitrary concentration for their confrontation with measurements., also exploiting TJ-II and LHD as a test-bed for validating physical mechanisms and modelling tools.


Go to the bibliography.