Turbulence simulations

Not only do collisions and the inhomogeneity of the magnetic field produce transport, but also turbulent fluctuations in the plasma do. The density and temperature gradients constitute free energy sources for the development of the so-called drift-wave turbulence [Horton-99], that can manifest due to the excitation of different types of instabilities: Ion temperature gradient (ITG) turbulence is considered to be responsible of most of the turbulent transport in the core of tokamaks and stellarators. The trapped electrons can generate the so-called trapped electron mode (TEM) instability even with flat temperature profiles, when a density gradient is present. In addition, an electron-scale microinstability, electron temperature gradient (ETG) mode, can also appear in presence of large electron temperature gradient [Horton-99].

In tokamaks, the ITG and TEM instabilities have already been extensively studied from the theoretical and numerical points of view and many comparisons with experiments have already been carried out. There are fewer simulations of ETG modes because of the huge computing resources required. In stellarators these instabilities have been less studied but several differences betwen stellarators and tokamaks have already emerged in this respect [Xanthopoulos-14, Helander-12]. In the case of ITG-driven turbulence, the unstable regions are more spatially localized as a consequence of the three-dimensional equilibrium magnetic field, while the level of instability appears to be similar to that found in tokamaks [Plunk-14]. For TEM, a similar localization can occur and some results indicate that the quasi-isodynamic configuration of W7-X could be almost immune to TEM modes, due to the non-correlated localization of trapped particle regions and bad curvature regions [Proll-12].

In the last years, we have carried out a characterization of electrostatic instabilities in the stellarator TJ-II by means of gyrokinetic (GK) simulations [Sánchez14]. Signatures of ITG, TEM and ETG have been found in different TJ-II plasmas. All the instabilities are spatially localized, the local magnetic shear and magnetic field line curvature being key magnitudes controlling the instability.

Zonal flows [Diamond-05] (that is, electrostatic potential perturbations that are constant on flux surfaces) are thought to play an important role in the self-regulation of turbulence in stellarators (as in tokamaks). Linear calculations have shown that zonal flows exhibit characteristic properties in stellarators that are different from those found in tokamaks [Sugama-05, Sugama-06, Mishchenko-08, Helander-11, Monreal-15]. We have studied the linear relaxation of zonal flows in different stellarator configurations by means of GK simulations and semi-analytical calculations [Sánchez-12, Sánchez-15, Monreal-15]. There are indications that the linear properties of zonal flow relaxation (residual zonal flow level and oscillations) can have an effect on the regulation of turbulent transport [Sugama-09, Nunami-12, Nunami-13, Xanthopoulos-11].

Equilibrium radial electric fields can also influence are turbulence regulation by modifying the linear instability and also the zonal flow linear damping. Due to the three- dimensionality of the stellarator configurations, the effect of a radial electric field on the linear instability, the zonal flows [Mishchenko-12] and the non-linear saturation level is more subtle than in tokamaks and its elucidation requires a detailed study. Different oscillations have been characterized in simulations of the linear relaxation of zonal flows in TJ-II [Sánchez-12]. Their dependence on the background radial electric field has been analyzed [Velasco-13]. Finally, some recent pellet injection experiments might provide empirical evidence for the existence of low-frequency oscillations in the relaxation of zonal flows.

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Impurity transport

Click on the figure to see the temporal evolution.

In magnetic confinement fusion, atomic species other than the fusion reactants (e.g. Deuterium and Tritium for the fuel mix envisaged for the first demonstration reactors) are termed “impurities”. The presence of even small concentration of impurities (especially high- Z ones) in the confinement volume has deleterious consequences on plasma performance (due to radiation power loss and fuel dilution). Furthermore, their accumulation in the core region can ultimately preclude the steady state operation of a fusion reactor. The achievement of long pulse operation without impurity accumulation in the high-heat flux divertor phase of W7-X (2019) is one of the milestones of the stellarator mission in the European fusion roadmap.

On the grounds of the standard neoclassical theory, such accumulation is expected to occur for medium and high-Z impurities, in the presence of a negative radial electric field (i.e. directed towards the core), see e.g. [Zurro-15, Zurro-15b]. A negative radial electric field is a natural condition of a high density fusion-grade plasma with strong ion-electron thermal coupling. This is particularly the case of stellarator-type reactors, for which no ion temperature screening effect is expected [Dinklage-13, García-Regaña-13] as in the tokamak configuration. Finally, the use of heavy species such as Tungsten as the divertor and first wall material is, to date, the preferred option to meet the requirements of heat exhaust, material erosion, high radiation fraction and Tritium retention.

In view of these facts, a robust strategy for the control of core impurity accumulation is needed as part of an integral solution to the several requirements to achieve magnetic confinement fusion. The two fundamental approaches are (1) to control the source of impurities from the divertor and plasma facing components by tailoring of the scrape-off layer (SOL) regime and (2) to act on the radial transport of impurities in the confined region. However, it should be noted that stringent conditions on SOL and core regimes are imposed by detachment and fusion performance respectively.

Plasma discharge scenarios with controlled core impurity concentration have been demonstrated in several devices. Central heating with microwaves in the electron and ion cyclotron frequencies has been shown to be instrumental for controlling impurities in tokamaks [Neu-02, Doyle-07]. In stellarators, experimental conditions with low impurity confinement time and very low impurity concentration levels in the core have been documented in W7-AS [McKormick-02] and LHD [Yoshinuma-09] respectively. However the physics behind these impurity control techniques and regimes remain poorly understood and their extrapolation to fusion-relevant conditions uncertain.

As a physical ingredient of core impurity transport, experimental and theoretical efforts have been invested in understanding impurity density variations along the magnetic field lines (i.e., variations on magnetic flux surfaces, often called “asymmetries”) [Ingesson- 00, Fülop-11, Viezzer-13, García-Regaña-13, Arévalo-14, Angioni-14, Rozhansky-15]. The specific dynamics of high charge and mass species can lead to large density asymmetries which can, in turn, alter their radial transport and accumulation. This is in particular the case when coupled to variations of electrostatic potential on flux surfaces, which have been recently diagnosed in the TJ-II stellarator [Alonso-14, Pedrosa-15]. Such coupling can potentially result in an outward convection of impurities like that observed in LHD’s “impurity hole” [Ida-09, Alonso-15]. Therefore, the investigation of the physical causes of those variations and their relationship with the magnetic geometry, radial electric field and collisionality is of great interest so as to assess their possible instrumentality for the much needed control of impurity accumulation in stellarators.

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Neoclassical transport and density control

Neoclassical transport is a fundamental aspect of stellarator plasma scenario development. As we have discussed, advanced stellarators are designed in order to minimize neoclassical radial transport. On the other hand, once designed, predictive 1D transport simulations based on neoclassical transport theory at the core, complemented with simple models for turbulent transport at the edge, allow for estimates of confinement time, power load to the walls, and fuelling and heating requirements of future reactors based on the stellarator concept. This way to proceed has been supported by a step-by-step systematic validation of the predictions of neoclassical theory with experimental results in a number of medium-sized stellarators (LHD, W7AS, TJ-II…) in regimes as reactor-relevant as possible [Beidler-11, Yokoyama-07, Dinklage-13, Tallents-14], in which our group has participated. Generally speaking, these simulations tend to show, with some exceptions, reasonable agreement between the experimental and the neoclassical particle and heat fluxes within the core region. This behaviour is caused by the unavoidable separation from perfect omnigeneity and the unfavorable temperature scaling of the radial energy fluxes with temperature [Wakatani-98].

One of the basic milestones of the Stellarator Mission in the European Fusion Roadmap is to demonstrate the validity of the design of advanced stellaratators (such as Wendelstein 7-X, to begin operation in Autumn 2015) based on neoclassical predictions. A relevant feature of these predictions is that all of their simulations rely on simplifications such us the monoenergetic approach or the local ansatz. Nevertheless, recent contributions [Satake-14,Velasco-14] have extended these calculations by investigating, for several discharges, differences between the results of local and monoenergetic neoclassical codes (e.g. DKES [Hirschman-86]) and non-local, non-monoenergetic neoclassical codes (FORTEC-3D [Satake-08]), which include in the calculation higher order terms of the drift- kinetic expansion. A preliminary result of these works is that, for the plasma core, neglecting these terms may lead to overestimate the neoclassical contribution to radial ion energy transport, and therefore to underestimate the turbulent contribution to it. The discharges analyzed so far suggest that the effect is larger for low collisionalities in large ripple devices [Satake-14, Velasco-14]. Additionally, flows parallel to the magnetic field give rise to a toroidal current (the bootstrap current) that may affect the magnetic structure at high pressure [Peeters-2000, Velasco-2011]. Since the design of the W7-X divertor (the part of the device responsible for particle and energy exhaust) relies on a very accurate tailoring of the magnetic field [Feng-2006], a study of the effect of the aforementioned higher-order terms of the drift-kinetic equation in the neoclassical prediction of bootstrap current is in order.

Out of the three phenomena that we have discussed above, bootstrap current and energy and particle radial transport, the latter has attracted much especific interest in the recent years. The reason is that core density control is a critical issue on the path towards the development of steady-state scenarios in 3-D magnetically confined devices. This is specially the case for the W7-X line, where coupling of neoclassical energy and particle transport in the core can lead to hollow density gradients, unstable pressure profiles, and hence premature termination of the discharge [Maassberg-99]. One of the strategies developed in order to mitigate this problem consists on pellet injection [Pégourié-07]. With this in mind, a compact pellet injector [McCarthy-08]. has been recently commisioned for the TJ-II stellarator [McCarthy-15], and has become part of a recently-established inter-machine comparative activity [McCarthy-15b]. The pellet injection system allows to inject frozen hydrogen pellets into magnetically confined plasmas for fuelling, for modifying the density profile and as an active diagnostic (the local magnetic field pitch can be determined or suprathermal electron populations can be located). It is a powerful tool for plasma fuelling since it permits delivering particles directly to the plasma core.

From the theoretical point of view, pellet ablation is a complex phenomenon, involving atomic processes and electrostatic or magnetic shielding. When a pellet arrives at the plasma, ablation starts. Sublimated hydrogen forms a neutral cloud that shields it from ambient plasma, in a self-regulated manner, reducing the ablation rate. The cloud expands and moves with the pellet, while atomic processes take place, until ionization occurs. It then experiences a Lorentz force that stops its transversal movement and the ionized cloud starts to follow the magnetic field lines. Accurate models describing ablation, able to determine ablation rate, penetration depth, particle deposition or fueling efficiency must take all these processes into account. These models, together with a similarly accurate description of particle transport, can shed light on our capabilities to control the density profile in large stellators and achieve long-pulse operation. There is ongoing research in this field at the stellarators LHD [Matsuyama-12, Yasuhara-14] and W7-X [Turkin-2013].

In spite of its importance, estimates of experimental particle transport remain inaccurate. The reason is that standard calculations (e.g. with ASTRA or TASK-3D transport suites) consist of calculating the particle source and then extracting the experimental particle flux from the particle balance equation. This kind of calculation has the drawback of an incomplete knowledge of the particle source: its precise calculation would require self- consistent modelling of plasma edge transport and plasma-wall interaction, which is still beyond the state of the art [Geiger-15]. Perturbative particle transport experiments can be devised to solve this limitation.

While particle and energy transport in the plasma core are collisional phenomena, both are determined by turbulence closer to the plasma edge [Dinklage-13]. Nevertheless, for non-quasisymmetric stellarators, radial transport of charge is still dominated by the neoclassical contribution if the gyrokinetic ordering holds [Calvo-13, Calvo-14, Calvo-15]. Thus, the ambipolarity condition of the neoclassical fluxes imposes a constraint between the kinetic profiles (density, temperatures and radial electric field) which is usually exploited in order to predict the radial electric field profile (and the associated ExB flow) for given plasma conditions, see e.g. [Arevalo-12, Velasco-12, Velasco-12b, Velasco-14, Velasco-14b]). There are situations [Dinklage-13] in which ambipolarity of the neoclassical fluxes does not provide an accurate estimate of the radial electric field. In those cases, comparison with the experiment allows for an estimate of the non-neoclassical contribution to the radial current.

Since there is experimental confirmation that turbulence can be regulated by sheared radial electric fields [Burrell-97], neoclassical simulations can provide information about the tendency of a magnetic configuration to access to scenarios of reduced turbulence and improved confinement. The relation between mean and fluctuating radial electric fields in connection with confinement transitions has been studied from both the experimental and theoretical point of view at TJ-II [Estrada-09,Velasco-14b], and similar diagnostic capabilities will be available at the stellarator W7-X.

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Transport in optimized stellarators

The three-dimensional geometry of the stellarator magnetic configuration (and therefore its increased number of degrees of freedom with respect to the axisymmetric configuration of the tokamak) can be used to generate the equilibrium magnetic field by means of external coils, instead of requiring a large plasma current to generate part of it, as in the tokamak. This prevents the existence of current-driven plasma instabilities and gives a conceptually more straightforward path towards steady-state operation.

However, the magnetic configuration of a stellarator has to be designed very carefully for it to have confinement properties comparable to those of an axisymmetric tokamak. In a generic stellarator, trapped particle orbits have non-zero secular radial drifts and they leave the device in a short time. The stellarator is called omnigeneous if the magnetic configuration is chosen so that the secular radial drifts vanish. Omnigeneity guarantees that the neoclassical transport level of the stellarator is similar to that in a tokamak.

The proof of Cary and Shasharina [Cary-97a, Cary-97b] for the existence of omnigeneous magnetic fields implies, at the end of the day, that exact omnigeneity requires non-analiticity. Let us explain this in more detail. As shown in [Cary-97a, Cary-97b], analytic omnigeneous magnetic fields coincide with the set of quasisymmetric magnetic fields [Boozer-83, Nührenberg-88]. To the virtues of omnigeneity, quasisymmetry adds the vanishing of neoclassical damping in the quasisymmetric direction. Therefore, in quasisymmetric stellarators larger flow velocities can be attained. In principle, this makes the stellarator plasma prone to develop large flow shear, that is known to reduce turbulent transport [Connor-04]. However, the quasisymmetry condition is incompatible with the magnetohydrodynamic equilibrium equations in the whole plasma [Garren-91], and the stellarator can be made quasisymmetric only in a limited radial region. The mathematical obstructions to achieve quasisymmetry do not exist for omnigeneity. This is why we said above that a necessary condition for exact omnigeneity is non-analiticity; specifically, the discontinuity of some derivatives of second or higher order.

In the last few years, we have focused on quasisymmetric stellarators. Since exact quasisymmetry is impossible to achieve even mathematically, the interesting question is to formally determine when a stellarator can be considered to be sufficiently close to quasisymmetry in practice. In a series of papers, and for different collisionality regimes and geometric properties of the deviations, we have answered this question and we have computed the effect on flow damping and on neoclassical transport of the unavoidable deviations from quasisymmetry [Calvo-13, Calvo-14, Calvo-15].

Although, in principle, omnigenous configurations exist, designing and aligning coils that create a magnetic field with discontinuous derivatives at certain points in space is probably technically impossible. The existing evidence leads one to conclude that even small deviations of omnigeneity are able to produce non-negligible enhancement of neoclassical transport. In this project, we propose to generalize the techniques developed in [Calvo-13, Calvo-14, Calvo-15] for stellarators close to quasisymmetry to the more general case of stellarators close to omnigeneity. A preliminary result on the freedom to design exactly omnigeneous magnetic fields has been published [Parra-15].

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