Modelling of electron transport and of sawtooth activity in tokamaks

Transport phenomena in tokamak plasmas strongly limit the particle and energy confinement and represent a crucial obstacle to controlled thermonuclear fusion. Within the vast framework of transport studies, three topics have been tackled in the present thesis: first, the computation of neoclassical transport coefficients for general axisymmetric equilibria and arbitrary collisionality regime; second, the analysis of the electron temperature behaviour and transport modelling of plasma discharges in the Tokamak à Configuration Variable (TCV) ; third, the modelling and simulation of the sawtooth activity with different plasma heating conditions. The work dedicated to neoclassical theory has been undertaken in order to first analytically identify a set of equations suited for implementation in existing Fokker-Planck codes. Modifications of these codes enabled us to compute the neoclassical transport coefficients considering different realistic magnetic equilibrium configurations and covering a large range of variation of three key parameters: aspect ratio, collisionality, and effective charge number. A comparison of the numerical results with an analytical limit has permitted the identification of two expressions for the trapped particle fraction, capable of encapsulating the geometrical effects and thus enabling each transport coefficient to be fitted with a single analytical function. This has allowed us to provide simple analytical formulae for al1 the neoclassical transport coefficients valid for arbitrary aspect ratio and collisionality in general realistic geometry. This work is particularly useful for a correct evaluation of the neoclassical contribution in tokamak scenarios with large bootstrap current fraction, or improved confinement regimes with low anomalous transport and for the determination of the plasma current density profile, since the plasma conductivity is usually assumed neoclassical. These results have been included in the plasma transport code PRETOR. This code has been further extended and applied to the simulation of electron transport in TCV. In simulating the electron temperature profile of Ohmic sawtoothing plasmas, the proper description of the current density profile and the sawtooth activity play the dominant role and not the specific transport model, provided that a single parameter in the model is adjusted to match the global plasma performance. In TCV discharges with electron cyclotron heating (ECH), the behaviour of the electron temperature exhibits some characteristics which have been recently observed to be common to several tokamaks. In particular, with central heating the electron temperature profile is stiff outside the power deposition region, that is the gradient scale length is independent of the heating power and essentially constant along the minor radius. With off-axis heating, transport is strongly reduced in the central region of the plasma, whereas a steep increase of the heat conductivity is observed at the power deposition location. Although the semi-empirical Rebut-Lallia-Watkins (RLW) transport model does not involve a critical gradient scale length, as the experimental observations would suggest, rather a critical electron temperature gradient, we have shown that it allows simulations which reproduce the described experimental features with very good agreement. Due to the relatively low toroidal magnetic field of TCV, the experimental temperature gradient with ECH exceeds by far the threshold included in the model. It can thus be stated that the parametric dependence of the electron heat conductivity of this transport model is adequate to reproduce the electron transport for plasma parameters in the operation domain of TCV. PRETOR, interfaced with the experimental data and the code TORAY-GA for the computation of the ECH source, has hence been used as a reliable tool for transport analysis and planning of new experiments. This has contributed to the identification of an improved central electron confinement (ICEC) regime in TCV, characterized by a precise heating scheme (strong electron cyclotron current drive in the counter-direction in the plasma center, and localized off-axis heating), with a specific time sequence. Transport simulations and investigations of this regime, in particular dedicated to the reconstruction of the current density profile during the high performance phase, have motivated further experiments which have confirmed numerical predictions. As a consequence, the magnetic shear reversa1 has been identified as the crucial ingredient for ICEC. The powerful ECH system in TCV does not allow only strong global current profile modifications but also local tailoring which has significant effects on sawtooth activity. A model introduced for the prediction of the sawtooth period in the proposed International Thermonuclear Experimental Reactor (ITER) has been extended to be applicable to Ohmic and ECH discharges in TCV. The model has been found in agreement with the experimental observations and thereby we were able to identify the physical mechanisms which make ECH capable of controlling the sawtooth period. The parameter dependence of the relevant stability threshold has been found consistent with dedicated experiments demonstrating the effects of localized heating and current drive on the sawtooth period. The simulations have pointed out the effects of the magnetic shear and of the pressure gradient at the q = 1 surface. Moreover, the most efficient heating location to stabilize the sawtooth period has been identified as located outside the q = 1 surface before the sawtooth crash. The same period model has been used for the simulation of the sawtooth period in recent discharges with neutral beam injection (NBI) in the Joint European Torus (JET), performed to assess the role of beam ions on sawtooth stabilization. With the inclusion of an analytical expression for the fast ion contribution to the internal kink potential energy, validated by the hybrid kinetic/MHD code NOVA-K, the simulations have been found in remarkable good agreement with the experimental observations. This work has demontrated the role of beam ions in sawtooth stabilization and validated the stability threshold for the resistive internal kink which was predicted to be the sawtooth crash trigger relevant for ITER operation.

Particle transport in tokamak plasmas

Alexey Zabolotskiy

The transport of particles in magnetically confined plasmas is of great importance for the development of fusion energy. It will determine techniques for fuelling, for controlling impurity concentrations and for the removal of the alpha particles produced by fusion reactions. The issues related to particle transport have received, until relatively recently, less attention than those regarding heat transport. Besides the greater experimental difficulty of measuring particle transport, this may explain why our understanding of this subject is still incomplete. The aim of this thesis is to document and tentatively interpret the experimental density profile behaviour in the TCV (Tokamak à Configuration Variable) and JET (Joint European Torus) tokamaks in the framework of different theoretical and semi-empirical models. The TCV tokamak is well suited to transport studies due to its extreme shaping capability, which allows the exploration of a wide range of different plasma conditions. This versatility is matched by the powerful and flexible electron cyclotron heating (ECH) system on TCV, which allows a control of the local power deposition profiles and current drive profiles. A study of particle transport on the JET tokamak has allowed us to compare the results of TCV to those of a much larger device and supplement the TCV study with the analysis in reactor relevant high confinement regime (H-mode). The experimental information was compiled into a database of density profiles in steady state, containing nearly 1000 samples for TCV and 600 samples for JET. The data analyzed covered a wide range of discharge conditions, including low confinement regime (L-mode) and H-mode discharges, ECH, ICH (ion cyclotron heating), LHCD (low hybrid), beam heated plasmas and include fully current drive discharges. The most relevant parameters which influence the density profiles were determined by regression. A detailed analysis of the particle sources showed that edge fuelling in TCV and JET cannot be responsible for density gradient in the plasma bulk, confirming the presence of particle convection or a 'pinch'. The existence of an anomalous pinch was unambiguously demonstrated both on JET and on TCV by the observation of peaked density profiles in stationary, fully relaxed, fully current driven discharges and hence in the absence of the neoclassical Ware pinch. An unexpected difference in the parameter dependencies was found in L-and H-modes. In TCV and JET, density gradient lengths (or profile peaking parameters) in L-mode were found to depend on magnetic shear with no dependence on collisionality. This lack of collisionality dependence in L- mode is inconsistent with current theoretical models. H-mode density profiles in JET, on the other hand, are clearly dependent on collisionality in agreement with theory and a prior observation on ASDEX-Upgrade, while exhibiting only a weak or no dependence on shear and temperature profiles. It was found that for TCV and JET, L mode density peaking can be interpreted as being due to turbulent equipartition, which assumes conservation of the magnetic moment and the longitudinal invariant during transport. The observation of a reduction of peaking in TCV with ECH supports drift wave turbulence theory, which predicts the appearance of outward particle convection, when trapped electron modes are destabilized. In JET H-mode, the weak secondary correlation of peaking with the electron-ion temperature ratio Te/Ti, may also be considered, at least quantitatively, as being supportive of drift wave turbulence theory.

EPFL2005

Modeling and control of the current density profile in tokamaks and its relation to electron transport

Costanza Zucca

The current density in tokamak plasmas strongly affects transport phenomena, therefore its understanding and control represent a crucial challenge for controlled thermonuclear fusion. Within the vast framework of tokamak studies, three topics have been tackled in the course of the present thesis: first, the modelling of the current density evolution in electron Internal Transport Barrier (eITB) discharges in the Tokamak à Configuration Variable (TCV); second, the study of current diffusion and inversion of electron transport properties observed during Swing Electron Cyclotron Current Drive (Swing ECCD) discharges in TCV; third, the analysis of the current density tailoring obtained by local ECCD driven by the improved EC system for sawtooth control and reverse shear scenarios in the International Thermonuclear Experimental Reactor (ITER). The work dedicated to the study of eITBs in TCV has been undertaken to identify which of the main parameters, directly related to the current density, played a relevant role in the confinement improvement created during these advanced scenarios. In this context, the current density has to be modeled, there being no measurement currently available on TCV. Since the Rebut-Lallia-Watkins (RLW) model has been validated on TCV ohmic heated plasmas, the corresponding scaling factor has often been used as a measure of improved confinement on TCV. The many interpretative simulations carried on different TCV discharges have shown that the thermal confinement improvement factor, HRLW, linearly increases with the absolute value of the minimum shear outside ρ > 0.3, ρ indicating a normalized radial coordinate. These investigations, performed with the transport code ASTRA, therefore confirmed a general observation, formulated through previous studies, that the formation of the transport barrier is correlated with the magnetic shear reversal. This was, indeed, found to be true in all cases studied, regardless of the different heating and current drive schemes employed. The increase of confinement with the negative magnetic shear was observed to be gradual, but constant, and did not depend on specific values of the safety factor. Therefore, the transition from standard to improved confinement appeared to be smooth, although it can be very fast. The flexible EC system in TCV allowed us to attain strong global confinement improvement to produce eITB regimes. It also permitted us to perform transport studies on plasmas characterized by low confinement, in which we modified the magnetic shear profile, locally, around the deposition location. For instance, alternate and periodic injection of co- and counter-ECCD within the same plasma discharge has been realized on TCV, while maintaining the same amount of total input EC power. Such a heating scheme has been the basis of Swing ECCD experiments, which were initially carried out using nearly on-axis EC deposition locations in the plasma, in order to maximize the EC power absorption, and therefore the magnetic shear variation. ASTRA, interfaced with the experimental data and the CQL3D code for the computation of the EC heating and current drive sources, has again been used as a reliable tool for transport analysis and planning of new experiments. The simulations have pointed out the effects of Swing ECCD on the magnetic shear and on the electron temperature profile around the radius at which the EC waves are absorbed. Both profiles turned out to be modulated at the same frequency as the frequency of the Swing ECCD. Moreover, the maximum magnetic shear variation has been observed to be independent of the transport models used for the simulations, therefore underlying the robustness of the modeling. Additionally, the numerical results have motivated further experiments with more off-axis EC deposition, which were found roughly in agreement with recent gyrokinetic predictions, according to which, at higher positive values of the magnetic shear, an inversion of the transport properties should occur. The aim of the study regarding the ITER project has been to analyse the capabilities of a possible variant of the EC system, recently proposed with the intent to optimize the combined action of the Upper and Equatorial EC Launchers and, therefore, to allow a broader operational domain for ITER. This variant will maintain the main goals for which the ITER EC system has originally been designed, namely the stabilization of Neoclassical Tearing Modes (NTMs) and of the sawtooth instability. This is a necessary feature that has been carefully analyzed in the present study. Besides allowing excellent performance in controlling NTMs and the sawtooth period, the suggested variant paves a way for further exploitation of EC waves for ITER. The present ITER base-line design has all EC launchers providing only co-ECCD. The performed numerical modeling has shown that the possibility to drive counter-ECCD with one of the three rows of equatorial mirrors offers greater control of the plasma current density. The counter-ECCD may also be balanced with co-ECCD to provide pure EC heating, with no net driven current. This would be an additional asset if EC waves were found to be needed to assist the L-H transition during plasma ramp-up. The overall decrease in co-ECCD, by turning one row to counter-ECCD, is estimated to be negligible, because the difference between full off-axis co-ECCD using all 20 MW from the Equatorial Launcher or co-ECCD driven by 2/3 from the Equatorial Launcher and 1/3 from the Upper Launcher is small. Therefore the latter analysis provides, in our opinion, a strong evidence of the substantial gain in flexibility if the suggested variant of the ITER EC system were accepted as the base-line design.

EPFL2009

Analysis of plasma dynamic response to modulated electron cyclotron heating in TCV tokamak

Ilya Pavlov

The need of durable, economically acceptable and safe energy sources continues to stimulate studies in a field of thermonuclear fusion. The most successful solution for controlled magnetic fusion is the tokamak. The improvement of tokamak performance depends on the optimization of pressure and current density spatial distributions which can be modified by means of an auxiliary heating and a current drive. Electron cyclotron heating, in particular is a very important tool for the study and control of basic physical processes governing plasma confinement and stability, particularly because it allows the injection of highly localized intense power. ECH power deposition location plays a crucial role in sawtooth control and suppression, it is also important for tearing mode stabilization, and for implementation of closed loop systems for automatic control/suppression of magnetohydrodynamic activity. A part of the ECH power can be modulated (MECH), and used to identify where the ECH power has been deposited, and can also be useful in the experimental analysis of the electron transport in general. Nevertheless, despite the goal of MECH being a diagnostic and analysis tool, MECH can couple to plasma oscillations, such as sawteeth. MECH-sawtooth phase coupling adds significant complications in ECH deposition location and transport analysis, in some cases making the interpretations of results misleading. This is why it is important to get an insight into the phenomenon of MECH-sawtooth interaction, and to establish the boundaries where conventional types of modulation analysis can be successfully implemented. This thesis presents the analysis and interpretation of perturbative MECH experiments performed in the TCV tokamak with particular attention paid to the non-linear phase coupling of heat waves. TCV is equipped with a very flexible and high power ECH system. Two independent ECH systems permit MECH to be deposited at two different spatial locations, with two different types of modulating signals. This set-up was used to study simultaneous propagation of heat waves induced by MECH in non-sawtoothing plasmas, and in discharges with sawtooth activity. A new analysis method for the characterization of the plasma non-linear dynamic response to modulated heating was developed on the basis of Higher Order Spectral Analysis (HOSA) technique. This method applied to signals from different diagnostics, such as electron cyclotron emission and soft X-ray measurements, was extensively used to quantitatively characterize the effect of nonlinear phase coupling. In sawtooth free discharges a detailed analysis of the propagation of heat waves demonstrated that their phase coupling is solely related to properties of heat sources. It was demonstrated that if two heat waves are induced by non-coupled power sources (multi-beam MECH) then no phase coupling occurs. In the opposite case, when a source of perturbation (MECH) contains coupled harmonics, the corresponding heat waves demonstrate phase coupling. It was shown that these coupled heat waves loose their phase coherence while propagating in plasma. The dissipation of phase coupling is due to different phase velocities of heat waves and their diffusive damping. The new type of ECH power modulation accompanied with bicoherence analysis was proposed as a candidate for a reliable identification of EC power deposition location in a case of high frequency and low modulation depth MECH, including multi-beam heating. This type of MECH can be particularly important for real time control applications. In cases when MECH is applied to sawtoothing plasmas a direct experimental evidence of MECH-sawtooth non-linear phase coupling has been demonstrated using HOSA techniques, in particular bispectrum and bicoherence profiles. The detailed analysis presented here demonstrates a direct proof of periodic modification of sawtooth behavior by modulated ECH. It was shown that a simple diffusive model for the perturbed electron temperature with MECH source term can not be used for transport analysis in the presence of MECH-sawtooth coupling. Self-consistent time dependant transport simulations including a sawtooth relaxation model is capable of properly reproducing the effect of MECH-sawtooth phase coupling and can be used for transport analysis. On the basis of these simulations, an hypothesis for MECH-sawtooth coupling based on periodic modifications of magnetic shear was proposed. Additional information provided by HOS analysis enabled a correct interpretation of the plasma dynamic response to modulated heating in the presence of MECH-sawtooth coupling.

EPFL2008