## CHAPTER B - B2 Development of Plasma Auxiliary Systems

### 2.1 Heating and current drive systems

### 2.1.1 Gyrotron interaction and cavity design

Background and Objectives: This activity addresses gyrotrons, which have been seen as the most promising configurations of high-power, high-frequency RF sources for Electron Cyclotron Resonance Heating and Current Drive. It is performed in close interaction with the leading European institutes in gyrotron studies and development and is continuously adjusting to the evolving needs. These needs encompass either the design and testing of gyrotrons for present and near-future fusion experiments, or the necessary advancements in theory and modelling in preparation for the design of the next generation of high-power gyrotrons, relevant to DEMO. The performed work refers both to designing suitable gyrotron cavities (conventional or coaxial) and to studying the fundamentals of the gyrotron interaction (also at higher cyclotron harmonics). Furthermore, pertinent numerical codes have been developed, which admit significant improvements and extensions. The codes have been integrated into the code-package EURIDICE for gyrotron design and simulation, developed at NTUA.

Work performed in year 2012 (in co-operation with KIT, CRPP, and UofLatvia):

(i) The effort on the implementation of an advanced integral boundary condition for the electric field profile in the self-consistent interaction code of the code-package EURIDICE was continued. The mathematical formulation of the advanced broadband boundary condition was reassessed and appropriate modifications were made, by which the problems in numerical stability encountered in the previous period were solved. Following that, the advanced boundary condition was successfully implemented in the code. The work on the boundary condition during this period was performed, in part, within the Call for Participation to the Task Agreement WP12-DAS-HCD-EC-04 (DEMO Design Assessment Studies, HCD, Electron Cyclotron, Numerical tool development), issued by PPP&T. An account of the work, also referring to progress in previous periods not reported in detail back then, can be found in ANNEX 21. In addition, further improvements and verification of EURIDICE were pursued. A summary is provided in ANNEX 22.

(ii) Studies on dynamic After-Cavity Interaction (ACI) in the 1 MW, 140 GHz gyrotron for W7-X were continued with the use of the advanced versions of EURIDICE. The results can be summarised as follows: (a) Single-mode simulations using the trajectory version of EURIDICE with single-frequency boundary condition (i.e. usual set-up) predict dynamic ACI of the operating TE_{28,8} mode at the frequency of ~132 GHz. This is in agreement with the results of the trajectory code SELFT (KIT). (b) The same simulations, but using the advanced broadband boundary condition, yield again dynamic ACI at about the same frequency. It is thus concluded that the artificial numerical reflection at the ACI frequency caused by the single-frequency boundary condition is not the reason for ACI appearance.

(c) Single-mode simulations using the quasi-PIC version of EURIDICE with single-frequency boundary condition do not predict any ACI. (d) Multi-mode (33 modes) start-up simulations with the trajectory version of EURIDICE with single-frequency boundary condition do not predict ACI of the TE_{28,8} mode but they show low-power parasitic excitation (~50 kW) of the TE_{26,9} mode at ~134 GHz at the up taper. The above results and discrepancies are under investigation, in parallel to the verification and assessment of the advanced versions of EURIDICE.

(iii) A preliminary study on the cavity design for a 240 GHz, 2 MW coaxial gyrotron for DEMO was performed. EURIDICE routine Design was used to identify possible operating modes obeying a series of physical and technological constraints, focusing on the lowest eigenvalue possible to reduce mode competition. It turned out that the lowest-order candidate is the TE_{48,29} mode (eigenvalue ~156). A preliminary coaxial cavity design was proposed and a suitable operating point was identified. Single-mode simulations showed that the TE_{48,29} mode is capable of 2.3 MW of generated power at 240 GHz, at a beam voltage of 87 kV, beam current 75 A, pitch factor 1.3, and magnetic field 9.7 T. The electronic efficiency is 35% and the Ohmic wall loading of the outer wall is 2.3 kW/cm^{2}, which is somewhat increased compared to the proposed limit of 2 kW/cm2. This increase, however, is considered acceptable for such an ambitious design. The next foreseen step is the verification of the above operating point with multi-mode simulations taking mode competition into account.

### 2.1.2 Mathematical modelling and numerical codes for gyrotron beam-tunnels and cavities

Background and Objectives: The parasitic oscillations in the gyrotron beam tunnel have been observed in present-day high-power gyrotrons (e.g. EU gyrotrons for W7-X), and is expected to be even more serious in future, as more powerful gyrotrons will operate at higher beam currents. Dielectric loading of the beam tunnel with lossy ceramics seems not to be sufficient to overcome the problem as long as the geometrical and physical properties of the whole structure are not considered in the design. This activity aims at realistic modelling of electromagnetic waves in dielectric loaded beam tunnels as also at providing basic design directions towards efficient suppression of parasitic oscillations. The modelling approach involves a semi-analytic treatment of the finite structure with outgoing-wave boundary conditions at its ends, in order to achieve the above goals. The gyrotron beam tunnel has a rich electromagnetic spectrum (especially in the presence of corrugated walls), part of which might resonate with the electron beam, as it is in transit to the gyrotron cavity. Such an interaction may have significant consequences, regarding the quality of the electron beam. For these reasons, this activity aims at the development of numerical codes, to calculate the frequency spectrum in typical gyrotron beam tunnel assemblies, with the prospect of eventually extending the codes to treat the electron beam self-consistently. In parallel, coaxial gyrotrons employ slotted cavities to facilitate mode selection. Such structures are typically calculated by employing the model of distributed impedance and therefore the calculations are limited to the domain of applicability of this model. This activity also aims at the development of numerical codes for the calculation of the frequency spectrum of slotted coaxial cavities, to allow performing calculations for cases which are beyond domain of validity of the aforementioned model.

Work performed in year 2012 (in co-operation with IHM/KIT, CRPP/EPFL, PSFC/MIT):

(i) A parametric study has been performed regarding the effect of the lossy ceramic material properties and the waveguide structure geometry on the parasitic modes in a stacked-type gyrotron beam tunnel. In this study the in-house numerical code Fishbone has been used for the numerical computations. From the numerical results it has been found that the depth and the dielectric permittivity of the ceramic material affect the parasitic oscillation in a similar manner. In addition, it has been seen that highly lossy materials do not necessarily lead to higher attenuation of the parasitic modes. In addition, studies for the collector sweeping coils behavior with respect to the frequency have been performed by using the CST Studio Suite. These studies have been performed in order to keep the heat dissipation of the collector’s surface within the technically acceptable limits. Two types of sweeping systems are currently used, the Parallel Frequency Sweeping System (PFSS) and the Transversal Frequency Sweeping System (TFSS). In the first case, the sweeping coils surround the collector creating a harmonically varying magnetic field, which is coaxial with the gyrotron’s static one. In the second case, the coils are distributed symmetrically around the collector, creating a rotating magnetic field that is transverse to the static one. In both cases the periodic variation of the magnetic field induces eddy currents in the conductive walls, reducing in this way the efficiency of the sweeping system. It was found that the attenuation observed due to the induced eddy currents depends not only on the material conductance, but also on the sweeping system. The magnetic field attenuation is accompanied by a phase shift, which has to be taken into account if a low frequency modulation is combined with the higher sweeping frequency. This phase shift can be described analytically for a wide frequency range (see ANNEX 23). In parallel, a new type of coaxial cavity with surface corrugation on both the inner and outer wall has been studied. The Spatial Harmonics Method (SHM) has been employed to find the characteristic equation for the eigenvalues of TE_mp modes for a cross section of such a cavity. This extra corrugation is expected to be an additional means to enhance the selectivity properties of coaxial gyrotron cavities. Such a cavity could be employed in the development of multi-MW gyrotrons at frequencies above 200 GHz (see ANNEX 24, PPP&T BS WP12-DAS-EC-05-01).

(ii) The work performed in previous years for rectangular waveguides has been extended to cylindrical ones. In particular, we have made the mathematical formalism of the Maxwell equations in cylindrical coordinates using the FDTD method and we have developed the corresponding numerical code. Special care has been given to overcome the singularity appearing for ρ = 0 in E_{ϕ}, E_{z} and H_{ρ} field update equations. An empty cylindrical waveguide was simulated and the numerical results for the TE11 mode are in agreement with those appeared in the literature (see ANNEX 25). Nevertheless, further tests and benchmarking of the numerical code are needed and they will be performed in 2013. Note that this activity aims to evolve to a tool for the calculation of the beam wave interaction in a gyrotron beam tunnel or cavity by the direct integration of the equations of motion.

(iii) The extension of the model and code developed the previous year to treat the beam-tunnel structure as an open resonator, was completed. An analytical continuation on the complex plane of the frequency was performed and the code implementing a complex root-finding algorithm was developed. The code was tested and first results about the cold electromagnetic properties of beam-tunnel sections were obtained (see ANNEX 26). An assessment study on the possible extensions of the model was also performed. It was found that in order to consider more general boundary conditions other than outgoing waves one has to reformulate the initial equations accounting for a general reflection coefficient that enters also in the Fourier integrals. This extension is necessary for considering also the varying character of the inner radius in the beam-tunnel. It was found that it may be possible to consider the step-wise tapering of the inner radius by dividing the hollow region of the beam-tunnel in many regions each having a total reflection at its right end. Then mode matching in the longitudinal direction should be applied, but this employs the matching of two regions with continuous spectrum. These two extensions will be scheduled for the future. Finally, the modelling of azimuthal indentations requires the expansion of the fields also in the azimuthal direction that increases the complexity of the model and code by one order of magnitude. Therefore it was decided that such a consideration should be left for the future with lower priority and for a separate code that will enable to study the effect of the indentations. The code was tested and first results about the cold electromagnetic properties of beam-tunnel sections were obtained (see ANNEX 26, PPP&T BS WP12-DAS-HCD-EC-04). An assessment study on the possible extensions of the model was also performed.

### 2.1.3 Amplification of the Gaussian rf beam provided by a gyrotron, by a sheet electron beam

Background and Objectives: Recent developments concerning research and design aspects of gyrotron resonators as the major high-power RF source for effective ECR heating in fusion reactors, revealed that an additional RF-generating mechanism should also be under consideration especially in view of even higher power sources (beyond 2 MW) required in next generation fusion reactors like DEMO. Accordingly, the proposed configuration consists of a sheet electron-beam bearing a considerably high current, in relatively low current density, which intercepts the RF beam of a gyrotron, resulting in a substantial power enhancement of the latter, while keeping the propagation characteristics of the input signal. Even though the conceptual design of the proposed configuration was aiming at high-power generating devices, its universal design and features suggest that applications in a various range of input parameters may also be feasible.

Work performed in 2012 (in cooperation with CRPP and FZK):

(i) In order to determine the presence as well as the effects –if any- of the recoil radiation to the electrons’ motion, we applied the Poynting’s theorem along and in a close vicinity to their paths. The simulations showed that the power mismatch is increasing the closer we get to the beamlets rather in a greater distant, probably due to far-field-based formulation of the corresponding EM radiation. This outcome implies that our calculations are relatively rough concerning minor radiation effects such as the recoil radiation, which cannot be accurately assessed under these approximations. The wave-particle interaction in a higher harmonic of the magnetostatic field, allowing the operation of the FSQOG using conventional magnet coils, was considered of minor importance for the time being and will be reconsidered eventually.

(ii) The dependence of the power-balance to the parallel velocity was eventually considered as part of the previous task, as the detailed calculations for the kinematic parameters adequately suffice to incorporate the parallel-velocity results (see ANNEX 27).

(iii) The effective wave-particle coupling occurring in FSQOG concept, is mainly due to the high current that is carried by the sheet-beam in a relatively low current-density. Given that, a thorough examination of the operation parameters revealed that the interaction is beneficial (and reliable) for a wide range of kinetic parameters, provided the fact that the input power is remaining to high levels (~MW). However, extensive simulations showed that even though the FSQOG could be easily exploited in low-power RF applications, the low-current approximation adopted in order to preserve the shape of the field (in order not to inherit the characteristics of the e-beam), keeps the efficiency of the interaction in relatively low-levels, making the implementation of such a down-scaled device practically doubtful.

(iv) The further development of the parallel instance of the code is postponed.

(v) Instead of the previous task, a new task was initiated with the aim of providing an additional formalism to express the equations of motion of the particles, and the interaction in general, based on analytical methods of Hamiltonian dynamics. Our aim is to express the kinetic parameters of the wave-particle interaction in an analytical fashion, based on the appropriate action-angle variables and the utilization of canonical perturbation theory. The significant advantage of such a method is the accuracy in determining the kinetic parameters and therefore the corresponding radiation field (see ANNEX 28).

### 2.3 Plasma fuelling

### 2.3.1 3-D pellet modelling

Background and Objectives: The 3-D pellet modelling activity is performed in collaboration with IPP Garching, and involves the development of a truly 3-dimensional resistive MHD code for pellet -plasma interactions. The long-term objective of this activity is to develop a 3-D pellet code for realistic tokamak geometries for: pellet-plasma interaction studies, pellet fuelling of magnetic fusion devices, and ignition of magnetically confined plasma with pellets. The code that is being developed is 3-D in Cartesian geometry and resistive MHD. The code computes the ablation rate of moving or stationary pellets self-consistently, by calculating the energy fluxes irradiating the surface of a pellet. Currently emphasis is given to the study (clarification) of the effect of the magnetic field strength on the ablation rate, because of differences in scaling laws (from experimental data) on the dependence of the pellet penetration depth on the magnetic field strength.

Work performed in 2012 (in co-operation with IPP-Garching):

(i) Stopping length calculations have been implemented in the 3-D resistive MHD pellet code. The stopping length calculations are performed as beams of energetic electrons (and ions) penetrating the plasma (and depositing energy) along magnetic field lines. The Maxwellian energy distribution of the incident particles is represented by five discrete energy groups. The penetrating plasma particles reach the surface of the pellet thus the ablation rate is calculated, self-consistently, by the energy deposited on the surface of the pellet by these energetic plasma particles. A number of scenario runs have been performed with the 3-D code with stationery pellets where the ablation rate is based on stopping length calculations, for different magnetic field strengths and different plasma parameters. These calculations show that the ablation rate increases with increasing the magnetic field strength (while other plasma parameters remain the same). Previous calculations where the ablation rate is calculated by the temperature fluxes irradiating the pellet also show that the ablation rate increases with increasing magnetic field strength. In order to understand the effect of the magnetic field on the ablation rate, studies with 3-D code on the confining radii of plasmoids (pellet clouds) are in progress.

### 2.4 Real Time Measurement and Control

### 2.4.1 Automatic control of MHD instabilities

Background and Objectives: Tokamak operation currently relies on rather simple control commands, whereas the demanding requirements in ITER performance suggest that a more sophisticated control logic may be required. The main target is the design of algorithms for the simulation of real-time control of plasma MHD instabilities based on modern control concepts. Our first approach involves the development of a state-space, closed-loop algorithm for the description of ECCD-based stabilization of NTMs, including response models for the diagnostic sensors and controller design based on stochastic, robust and/or intelligent control tools. An accompanying task is to benchmark the established system identification methods on the accurate prediction of the (known as modeled) system dynamics.

Work performed in 2012 (in cooperation with TEIPIR):

(i) In this period, we continued the development of the algorithm for the feedback control stabilization of NTMs, with ECRH/ECCD as actuators and controller design based on modern control techniques. We have completed a first hands-on model of the block for the diagnostic sensors, based on a linear first-order system with dead-time, transition time and steady-state error within the known prescriptions for ITER diagnostics, and have finished the development of the block for the ECCD actuator with the asymptotic computation of the inverse wave propagation from the desired deposition point to the launch point. Also, the algorithm for the subsystem of the wave launcher, which involves the steerable mirror that guides the wave beam into the plasma and the motor that controls its rotational motions, was updated importantly, including among others cascaded control, friction compensation and AC servomotor modeling. The studies will be continued with the construction of a Simulink model for the feedback control of NTMs with the development of an upgraded version of the diagnostics block, based on the specific physics of the relevant processes, and of the controller design of the outermost loop involving prediction of the island dynamics.

Last Updated (Tuesday, 25 February 2014 10:32)