## CHAPTER B - B2 Development of Plasma Auxiliary Systems

## 2.1 Heating and current drive systems

### 2.1.1 Investigation of new concepts for high power microwave generation for ECRH: Application of sheet e-beam to increase output power of gyrotrons

Background and Objectives: Gyrotron oscillators are considered to be the most effective devices capable to deliver coherent millimetre and sub-millimetre radiation of approximately 2 MW in the range of 170 GHz, which is required for electron cyclotron heating in fusion plasma applications like ITER. In order to increase the output power of a conventional gyrotron, an alternative configuration very similar to the quasi-optical gyrotron (QOG) is proposed, in which a sheet electron beam immersed in a magnetostatic field intersects perpendicularly the rf beam produced by the gyrotron. As a result, the initial Gaussian-shaped output of the gyrotron is substantially amplified to the power levels of several MW’s that are required for a successful and efficient fusion reaction.

Work performed in year 2008 (in co-operation with CRPP and FZK):

(i) There has been extensive effort to test and benchmark both the efficiency and the speed of the simulation code, as well as to obtain accurate and comprehensive results (see **Annex 18**). Based on these numerical results, it was attempted to calculate the radiation fields (considering first the calculation of the corresponding potentials) using a standard model for the perturbed electron current flow (**Annex 19**).

(ii) The results of task (i), as well as those obtained during the previous annual periods, have been assessed to evaluate the potential of the interaction, primarily as regards its implementation for the generation of the high power required for Tokamak heating and current drive with a smaller number of units. The corresponding technological requirements on the emitter, the generation of the magnetic fields, the collector and the output window are discussed in **Annex 20**.

### 2.1.2 Gyrotron interaction and cavity design

Background and Objectives: This activity addresses both hollow-cavity and coaxial-cavity 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 European gyrotron development programme and is continuously adjusting to the evolving needs. In addition, harmonic interactions are also studied in this activity, for the purpose of producing high frequency at reduced magnetic field requirements. The work performed refers both to designing suitable cavities and to studying the fundamentals of the interaction. Furthermore, pertinent numerical codes have been developed, which admit significant improvements.

Work performed in year 2008 (in co-operation with FZK, CRPP and UofLatvia/TEKES):

(i) A PIC-like version of the existing parallelised, fixed-field, interaction code has been developed. In this version, a different modelling of the electron beam is employed (we call it “filled-cavity”), compared to the usual “electron ensemble” modelling. This alternative modelling is closer to reality, ensuring correct time averaging, which is important in the case of the modes that are radial satellites of a dominant mode (where azimuthal averaging has no effect). A study of the parasitic excitation of radial satellites, using the new version of the code, has shown that the usual “electron ensemble” modelling can give erroneous results if the time step is not small enough. It turns out that the radial satellites are indeed excited, but at lower power levels (and thus with less influence on the dominant mode) than those predicted by simulations with the “electron ensemble” modelling using the usual time step. Details on the two types of the electron beam modelling and on the excitation of radial satellites can be found in **Annex 21**.

(ii) Further improvement of the self-consistent subroutine for steady-state, single-mode operation, developed in the previous period, has been carried out. A good agreement of its results with those yielded by the corresponding FZK code, SELFC, was achieved. Furthermore, extension of the self-consistency, regarding the rf field profile, to the time-dependent, multi-mode part of the interaction code has also taken place. The new self-consistent code and all the previously developed interaction codes and designing tools have been integrated into the code package EURIDICE developed at NTUA. Parts of EURIDICE have been installed in the cluster “Pleiades 2” at CRPP. In addition, EURIDICE has been partially integrated into the code package CAVITY, existing at FZK. Comparisons between the new self-consistent code and the well-established FZK interaction code SELFT in single-mode calculations have shown excellent agreement. In similar calculations, EURIDICE results also agreed with those of the self-consistent code COAXIAL, existing at UofLatvia/TEKES. Further improvements of the self-consistent code and benchmarking in multi-mode calculations are foreseen for the next period. Regarding code extensions, the possibility of sweeping over the magnetic field was implemented both in the self-consistent code and in the code for the calculation of the starting currents of the modes. In addition, the self-consistent code has incorporated several options regarding the employed carrier frequency (i. e. the assumed frequency of the fast oscillations) because, as shown by SELFT, the results are sensitive to this option. An investigation on this is planned for the next period.

(iii) The work on the selection of the operating mode and the operating point for the 170 GHz, 1 MW conventional gyrotron (EU fallback solution for ITER) has been continued from the previous period. The TE32,9 mode appears to be the most promising candidate and an appropriate cavity design has been obtained. Details on these results (including also the work in the previous period, not reported extensively then) are presented in **Annex 22**. Further investigations on TE32,9 and also on TE34,10 (possibly capable of more power), using the new self-consistent code, are planned for the next period.

(iv) The NTUA interaction codes have been used to support the experiments with the EU 1st prototype 170 GHz, 2 MW coaxial gyrotron for ITER, existing at CRPP. Several operating points were simulated in single-mode, and information on what operating parameters should be tried was provided. Although the trend of the power variations was the same in both the simulations and the experiment, the calculated power values were initially about 20 % higher than the measured ones, a fact attributed to the uncertainty in the measured power, in the actual beam-alpha, and in the velocity spread. When the self-consistent code was provided with better estimations for these values the agreement was satisfactory. An even better agreement between single-mode, self-consistent calculations and experiment, was achieved with results from the pre-prototype 170 GHz, 2 MW coaxial gyrotron for ITER, existing at FZK, where the actual beam parameters and losses were estimated more accurately. Comparisons of multi-mode simulations with experimental results are foreseen for the next period.

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

Background and Objectives: The gyrotron beam tunnel, whether cylindrical or coaxial, 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, as regards the quality of the electron beam, even if no substantial energy exchange takes place. (Energy spread is typically proportional to the small quantity of the normalised field amplitude, whereas energy exchange is proportional to the square of it.) 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 interaction 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 2008 (in co-operation with CRPP, FZK and UofLatvia/TEKES-HUT):

(i) Beam loading for the hybrid modes in the coaxial gyrotron beam-tunnel numerical code: The mathematical formulation for the beam loading for the hybrid waves in a coaxial gyrotron beam-tunnel has been completed and the corresponding numerical code (CoaxBT) has been developed. Numerical tests for limiting cases of coaxial geometries as well as cold and hot structures have been performed to check this code and to ensure the validity of the results obtained. In particular, the results obtained for a cold smooth coaxial waveguide structure (i.e., without beam loading and corrugations) are identical with those found in the literature. In addition, the numerical results for a cold corrugated coaxial structure are reasonable and comparable with those for a conventional one (**Annex 23**). Finally, the numerical results for the hybrid modes with m = 0 are identical with those derived for the simpler cases of TE and TM modes. The numerical tests and runs will be continued in 2009.

(ii) Optimisation and test runs of the codes developed in the previous years for the conventional and coaxial gyrotron beam tunnels: An updated version of the mathematical formulation of the beam-wave interaction in gyrotron beam-tunnels for the cases of hybrid waves in the small-signal regime has been developed (**Annex 24**). The analysis is based on the linearised Vlasov equation in order to investigate any possible parasitic oscillation that may be excited from the noise. This updated formulation has been used in the code and checked. In addition, test runs and optimisations have been performed mainly for the conventional gyrotron beam tunnels. Several improvements of the corresponding code have been included, whereas our numerical results are in a good agreement with those reported in the literature (for BWOs and TWTs). In addition, after interaction with FZK it has been decided to extend this task and investigate the appearance of parasitic modes in the beam tunnel of the 140 GHz gyrotron for W7-X. Indeed, for a simplified model of the beam tunnel geometry the numerical results show possible parasitic oscillations in the frequency range 118-130 GHz, which are very close in frequency to those observed experimentally at FZK. It has been found that the parasitic modes are similar to TE0,11 and TE0,13 of the smooth waveguide. Furthermore, for a simplified model of the beam-tunnel of the 90 GHz gyrotron at CRPP, Lausanne, axisymmetric (TE) parasitic modes have been found. These modes could not be damped by the dielectric losses of the corrugation, since they are slow-wave modes and are mainly concentrated close to the beam radial position (Annex 25). This subtask will be continued in 2009, since parasitic oscillations in beam-tunnels is a very important issue for the gyrotron operation.

(iii) Identification of parasitic modes in the beam tunnel of conventional 170 GHz 1MW gyrotron under design for ITER: Since the details of the beam tunnel for the 170 GHz, 1 MW gyrotron (EU backup solution for ITER) are not known yet, only numerical tests for arbitrary beam tunnel geometries have been performed to check the performance of the code at that frequency range. Moreover, new normalisation procedures have been included in the code in order to avoid any possible overflow problems. This subtask will be continued in 2009, as soon as the details of the beam-tunnel geometry become available.

(iv) Extension of the numerical codes developed in the previous years to include other types of corrugation: The mathematical formulation to study the dispersion properties of TM modes in a circular waveguide with surface corrugation, which is described by a periodic function R(z), had been developed in 2007 as well as the corresponding numerical code. In 2008, several cases of BWOs have been considered and it has been confirmed that the method is limited to corrugation profiles satisfying the Rayleigh criterion, which describes the physical restriction due to the appearance of locked oscillations in the grooves of the rippled wall. In order to study complex profiles, such as an iris-loaded cylindrical waveguide with smoothed parts, the formulation has been combined with the mode matching technique. The numerical results of the extended method are in excellent agreement with the finite elements codes SUPERFISH and HFSS [I. G. Tigelis, J.-Y. Raguin, Z. C. Ioannidis, G. P. Latsas, and A. J. Amditis, International Journal of Infrared and Millimeter Waves 29, 432 (2008)]. Furthermore, this method has been also used to study the characteristics of TE modes in such structures. From the numerical results it has been found that the spectrum of TE modes remains unchangeable when the corrugation depth is increased, contrary to the TM modes, where it changes significantly. Finally, our results are in a very good agreement with those obtained by other codes (**Annex 26**).

(v) Development of the new code for the beam wave interaction in a gyrotron beam tunnel (conventional or/and coaxial) by direct numerical integration of the equations of motion: This subtask has not started yet due to the extended effort given on the high-priority subtasks (i) and (ii) as well as (2.1.2).

### 2.1.4. Chaotic and Hamiltonian electron dynamics in gyrotrons

Background and Objectives: The main objective of this activity is to analyse complex electron dynamics in gyrotron resonators in order to provide information about efficient operation of gyrotron devices. The analysis and the methods utilised are within the context of the Hamiltonian formalism, including phase space analysis, Canonical Perturbation Theory (CPT) and symplectic integration schemes.

Work performed in year 2008 (in co-operation with UofLatvia/TEKES-HUT):

(i) Analytical results for the calculation of phase-averaged quantities, based on higher-order perturbation theory, have been obtained up to fourth-order accuracy with respect to the beam to rf coupling factor.

(ii) Specific quantities of physical interest describing the collective electron behaviour have been considered for analytical calculation and compared with numerical results in order to investigate the domain of validity of the analytical results.

(iii) Analysis of electron interaction process in the course of gyrotron switching from one mode to another has been initiated. This analysis is based on the use of the Hamiltonian formalism that allows one to construct Poincaré plots for different instants of switching time. The study is carried out for a 170 GHz MW-class gyrotron for ITER. More details can be found in **Annex 27**. (This is a new task, added in view of the recently increased interest on mode competition for the gyrotron designed for ITER and is performed also in co-operation with Institute for Research in Electronics and Applied Physics, University of Maryland.)

## 2.2 Plasma diagnostics

### 2.2.1 Numerical simulations for fusion electrodynamic systems

Background and Objectives: TORPEX is a toroidal device, in operation (at CRPP, Lausanne) since March 2003, which aims at addressing, inter alia, (a) the relative contribution to the cross-field flux from correlated density and potential fluctuations, associated with unstable modes, or with isolated intermittent events and (b) the modes most relevant for transport and their relation to the specific configuration and plasma parameters of different devices. Microwaves are injected into the (toroidal) vacuum chamber from the side by an appropriate rectangular waveguide, with the waves being in the ordinary mode (O-mode) polarisation at the output of the waveguide. In addition, a transition from rectangular to circular cross-section is used to match the waveguide to the vacuum chamber. Since no focusing elements are present, microwaves are actually injected into the chamber with a mixed polarisation, which can be represented as a superposition of O- and X-mode. The objectives of this activity are the modelling and the numerical simulation of the corresponding electromagnetic problem, i.e., the calculation of the spectrum of electromagnetic waves, which are excited in the toroidal vacuum chamber, as well as the influence of the transition to the field properties. Furthermore, optimal transition configurations, which minimise the reflection coefficient at the excitation port, will be searched.

Work performed in 2008 (in co-operation with CRPP):

(i) The calculation of the full spectrum of waves in the whole toroidal waveguide structure has been unfeasible, since it is overmoded and we have faced memory problems during the numerical simulations. We have also modelled several simplified cases of the whole structure (i.e., simple cylindrical waveguide, section of a torus), but even in those simplified geometries the spectrum has not been calculated, because of the large dimensions of the system compared to the wavelength (overmoded structures). Finally, we have focused our efforts on the following two cases: (A) a cylindrical waveguide with isotropic plasma with or without taking into account the collisions and (B) a cylindrical waveguide with plasma, whose frequency is treated as a function of the density of the charged particles being described by a Maxwellian distribution centred at the origin, and the cross-section of the waveguide is discretised in areas, each one with different density and consequently different plasma frequency and relative dielectric permittivity. The results of these two cases are given in (**Annex 28**).

(ii) For Case (A), it has been found that the reflection coefficient at the excitation port is relatively high with or without taking into account the collisions. On the other hand, for Case (B), the return losses are significantly small without the two transitions, while the presence of these transitions (with dimensions equal to those of the exact geometry) results to a significant increase of the reflection coefficient.

(iii) The electric field distribution for Case (A) is not homogeneous inside the cylindrical waveguide, but it is mainly concentrated close to the centre of the circular cross-section, with or without taking into account the collisions. For Case (B), it has been seen that electric field distribution is not homogeneous, is located mainly in the half part of the circular waveguide near to the excitation port. The presence of the two transitions results to a decrease of the maximum of the electric field magnitude as well as a modification of the field profile.

### 2.2.2 Physical Properties of materials to be used in plasma diagnostics (NCSR“D”)

Background and Objectives: To assess and characterize the magnetic, electrical, optical and structural properties of materials considered for plasma diagnostics.

Work performed in 2008:

There have been discussions with the Diagnostics Topical Group on possible involvement in assessment of materials and characterisation needs (magnetic, electrical, optical, structural) in connection with plasma diagnostics.

## 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/Greifswald, and involves the development of multi-dimensional resistive MHD codes. The long-term objectives of this activity are to develop multi-dimensional pellet codes, and possibly a 3-D resistive MHD pellet code, for pellet-plasma interaction studies, pellet fuelling of magnetic fusion devices, and ignition of magnetically confined plasma with pellets. The mid-term objectives are to develop a single code which reproduces all relevant pellet ablation characteristics, i. e., (a) ablation rates and pellet penetration depths, and (b) radiation patterns produced by ablatant clouds and particularly investigation of visible striations aligned with B|| direction, and grad(B)-induced drift phenomena. In order to achieve the above objectives it will be necessary to couple the pellet code with transport codes and equilibrium codes.

Work performed in 2008 (in co-operation with IPP-Garching/Greifswald):

(i) In the 2-D+1 code with single Lagrangian cell in the z-direction, prior to performing the Lagrangian step, the plasma parameters were averaged with respect to the fixed length in the z-direction that the 2D MHD takes place and the total length of the Lagrangian cell, in the z-direction [P. Lalousis, L. L. Lengyel, R. Schneider, Plasma Phys. Control. Fusion 50 08500 (2008)]. In going into more accurate expansion in the z-direction, multi-cells, one Eulerian (in z-direction) and others Lagrangian, are being coupled to the 2-D grid, replacing the single-cell approximation. The temperature diffusion equation, which is one dimensional in the z-direction, has been implemented for every cell of the Eulerian grid, and tested. Spitzer thermal conductivities with free flux limiter are used.

(ii) On a three-dimensional Eulerian grid (of the 2-D+1 code. with one Eulerian cell in the z-direction), numerical modules have been developed which compute the various fluxes (for particles, momentum, and energy) in the interfaces of the 3-D cells in the z-direction. At these interfaces a number of Lagrangian cells are generated in the z-direction. This generation of Lagrangian cells, is a kind of automatic mesh refinement. The criterion for generating these cells is based on the mass of the cell compared to the mass of the neighbouring cell in the z-direction. The ablated particles are deposited only on the Eulerian cells, so only the first Lagrangian cell which is attached to the Eulerian cell is spitted based on the mass criterion. Computational modules of calculating the energy fluxes, irradiating a spherical pellet, thus calculating the ablation rate, have been implemented. Further improvements to the splitting of the first Lagrangian cells are necessary.

(iii) The temperature diffusion equation, and the magnetic field equations for the two components of B (Br, Bz) have been implemented in the 2-D MHD cylindrical code (see **Annex 29**).

Last Updated (Sunday, 03 April 2011 14:13)