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 recently integrated into the code-package EURIDICE for gyrotron design and simulation, developed at NTUA.

Work performed in year 2010 (in co-operation with KIT, CRPP, and UofLatvia):

1. This milestone has been included in the F4E grant agreement F4E-2009-GRT-049 and therefore the associated man-power was excluded from CoA.
2. Additional improvements of EURIDICE, regarding tapered magnetic field, RF reflections, and RF space charge have been included in the F4E grant agreement F4E-2009-GRT-049 and therefore the associated man-power was excluded from CoA. In the frame of CoA, further verification of EURIDICE through comparisons with existing codes was possible. A brief report is provided in Annex 17. A very important new finding was the prediction of dynamic After-Cavity Interaction at the 140 GHz, 1 MW gyrotron for W7-X. Due to the extended effort devoted to the issues related to the above new finding and due to a successful work-around [see (iii)], the planned improvement of the surface impedance model employed for coaxial cavities was not performed.
3. The new single-mode, fixed-field code (developed in the previous period), which simulates the interaction of the electron beam with a TE, TM or hybrid mode (close to or far from cut-off) in an axially varying magnetostatic field, was further tested and improved. The multi-mode, self-consistent code in the code-package EURIDICE, simulating the interaction of the electron beam with TE modes close to cut-off, was extended to take into account the influence of an axially varying magnetostatic field on the electron parallel and transverse velocities. The extension was verified by comparison of EURIDICE with the codes Ariadne++ and TWANG, existing at CRPP. Comparisons with the same codes revealed that the influence of the transverse RF magnetic field on the electron motion is relatively small even for modes not so close to cut-off. The planned extension of EURIDICE to include this influence was thus given lower priority and was postponed. In order to further increase the relevance of EURIDICE to interaction simulations at the beam-tunnel, two more topics were considered: First, axially dependent ohmic losses were implemented. It turned out that, in a good approximation, this was possible by using the concept of a “local” ohmic quality factor instead of proceeding to a much more demanding improvement of the surface impedance model, as was initially planned. This implementation offers additional means of obtaining results relevant to the beam tunnel, provided that the beam tunnel corrugations can be approximately described by a proper surface impedance. Such a description is under investigation. Second, the boundary condition for the RF field, as employed in EURIDICE, was discussed because its applicability becomes questionable as the taper of the geometry increases. (This can be the case at the beam tunnel). A more appropriate boundary condition, based on Airy functions, was identified. Work on formulating the new boundary condition in a way compatible with the code formalism was initiated. (In co-operation also with PSFC/MIT.)
4. Feasibility studies on the implementation of a symplectic mapping for electron orbit calculations have been initiated. The respective mappings have already been constructed in terms of Hamiltonian perturbation methods. As an unperturbed (zero-order) system either the free electron motion (in the absence of RF fields) or the electron motion in a plane (sinusoidal) RF field can be used. The latter can have the parameters of the actual RF wave close to the output of the cavity. The implementation and the coupling of new routines to existing codes have been postponed due to occupation of the available manpower with the higher priority tasks (i)-(iii).
5. Studies on the self-consistent Hamiltonian system describing both electron beam and RF modes dynamics were continued and theoretical results for a simplified model have been obtained.

2.1.2 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 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. 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 may be not 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 a basic physical understanding of the absorption of electromagnetic waves in dielectric loaded beam tunnels as also at providing basic design directions towards efficient suppression of parasitic oscillations. The methods used involve alternative electromagnetic modelling, reasonable approximations for realistic beam tunnels and analytic treatment of the problem in order to achieve the above goals.

Work performed in year 2010 (in co-operation with KIT, CRPP/EPFL, PSFC/MIT):

1. The numerical code Fishbone developed in the previous years to consider parasitic oscillations in conventional beam-tunnel geometries has been used to explain the parasitic oscillations observed in the W7-X gyrotron. Furthermore, a parametric study on the excited parasitic modes in a gyrotron beam tunnel was performed. We focused our attention to the effect of the dielectric material as well as of the slot geometry on the growth rate of the mode (see Annex 18).
   In addition, the numerical code CoaxBT developed the previous years for studying parasitic oscillations of TM, TE and hybrid modes (m = 1, 2,…) in coaxial gyrotron beam-tunnels for the case of periodic and non-periodic surface corrugation profiles was checked and optimised in order to be faster. For this purpose an adapted type of search was added in order to study only the area (complex  and axial wave number k_z), where the parasitic oscillations take place and not all the possible dispersion curves. This means that we are looking for roots close to the resonance condition  = / + k_zv_z for the transverse interaction or  = k_zv_z for the axial one. Extended test runs were performed.
   Moreover, in order for the two codes (for conventional and coaxial gyrotron beam tunnels) to have the same features and capabilities, we decided to include in the latter code the capability of the calculation of the cold structure properties (i.e., frequency spectrum and Ohmic losses). The mathematical formulation and the corresponding numerical code for a cold coaxial gyrotron beam tunnel were made. In the analysis, the beam-wave interaction was omitted, whereas the dielectric material in the grooves was treated as being without losses and the calculation of Ohmic losses was performed by the approximation P_ohm=2(_i/_r)W_eout, where  and W_eout are the frequency and the total electric energy inside the grooves, respectively (Annex 19).
   In view of further verification of the codes and of treatment of more complex geometries, we have also started simulations with the commercial software CST Studio Suite, which is available. Up to now, several attempts have been made in order to simulate the gun geometry as well as the whole beam-tunnel geometry. Nevertheless, problems with the mesh properties and the boundary conditions of the structure have prevented us from proceeding faster with the simulations.
   Finally, the analysis of hybrid modes in a circular waveguide with a surface corrugation described by a periodic function R(z) was not made during 2010, because the personnel was involved in other more urgent activities of the work programme as well as the F4E grant F4E-2009-GRT-049. For the same reason, the code developed for TM and TE modes in circular waveguide with a surface corrugation described by a periodic function R(z) was not checked and studies on geometries with inner radius of dielectric larger than that of the metallic walls as well as with smoothing at the edges of the corrugation, were not performed.
2. Regarding coaxial cavities with corrugated insert, several investigations were performed aiming mainly at understanding the way in which the spatial harmonics contribute to the reformation of the eigenvalue spectrum. After significant effort, a new stricter criterion was found for the cases where the spatial harmonics can be neglected and the simpler surface impedance model (SIM) can be applied safely. It was also verified that the usual SIM application criterion N > 2m can be insufficient in some cases, even for cavities with a large number of insert corrugations, working at high order modes (Annex 20).
3. Based on lengthy discussions with KIT, CRPP, and MIT, it was decided to stop this activity because the current mathematical formalism (and the imposed assumptions) of the Fishbone and CoaxBT codes cannot be extended to include axially varying magnetostatic field and thus we have to start from scratch to develop such a code. Unfortunately, there is no available personnel in the research unit to undertake such a development, which would need at least 2-3 years. In this context, it also becomes questionable whether the effort invested in such a development will be justified by the achieved results. The investigations on simulating the beam-wave interaction at the beam tunnel in the presence of axially varying magnetostatic field were focused, for the moment, on the pertinent extension of the EURIDICE code-package [see task 2.1.1(iii)].
4. Hands-on experience has been obtained with a full-wave numerical code (developed earlier by Prof. J. Tsalamengas of NTUA), which studies the reflection/absorption characteristics of a plane wave, incident on a plane surface containing rectangular grooves loaded with absorbing material. Numerical results indicated a direct relation between the absorption efficiency of the grooves of the plane surface and the quality factor of the cylindrical corrugated beam tunnel, as obtained by the code FISHBONE. Extensive simulations with the full wave planar code revealed also a strong dependence of the absorption efficiency on geometric and physical properties of the grooves. At the same time, the planar geometry approximation was found to be valid for the high order modes that concern the beam tunnel. Consequently, the task of initiating a new full-wave code for the reflection/absorption characteristics related to a cylindrical geometry (which would require cumbersome analysis, important numerical effort and is partially covered by existing full-wave codes like FISHBONE) was reassessed. It was decided that, in order to identify the conditions of maximising the absorption on the planar geometry and transfer them to the cylindrical beam tunnel structure, an alternative approach should been followed. In particular, a fully analytic simplified model based on the planar geometry and the corresponding computer code were developed (see Annex 21). An extensive testing of the code has been initiated and possible improvements to the model will be implemented in the future.

2.1.3. Amplification of a Gaussian rf beam provided by a gyrotron via its interaction with a sheet electron beam

Background and Objectives: In view of the needs of the fusion reactors beyond ITER, a new configuration is considered in order to assess the feasibility of amplifying a high power RF beam beyond 2 MW, which is currently the maximum achievable power by a single gyrotron. This conceptual design, which involves a sheet electron beam drifting along a magnetostatic field and intersecting perpendicularly with the output RF beam produced by a gyrotron, resembles the initial concept of the Quasi-Optical Gyrotron featuring high kinetic energy and large volume to deploy the electron beam. However, it operates with a single-mode propagating wave provided by a gyrotron, and supports a large amount of current in a relatively low current density. As a result, the initial Gaussian-shaped output of the gyrotron is substantially amplified to the power level of several MWs that is required for a more efficient fusion reaction.

Work performed in 2010 (in co-operation with CRPP and KIT):

1. The current distribution was approached by a semi-empirical law taking into account suitable expressions in order to include near-field approximations. The radiation field was obtained making use of the corresponding current expressions, selecting optimum values of the induced terms in order to closely match with the numerical obtained results (see Annex 22).
2. The pattern of the radiation field was determined for numerous interaction parameters (e.g., the energy or the current of the electron beam) and global conclusions were reached regarding its Gaussian cross-section. The numerical calculations were extended in order to incorporate the corresponding power-flow of the radiation field, as well as to determine the energy balance and the power gain for each set of interaction parameters. The obtained results exhibit the most efficient cases and yet explore the limits of the proposed configurations (Annex 23).

2.2 Plasma diagnostics

2.2.1 Diagnostic methods for the measurement of electron temperature

Background and Objectives: Systematic discrepancies between the temperature profiles measured by the Electron Cyclotron Emission and Thomson Scattering diagnostics have been observed in various JET experiments with high temperature plasmas with strong auxiliary heating. The objective of this work is the study and understanding of this discrepancy in temperature measurements at JET and how the discrepancy appears in the Electron Cyclotron Emission spectrum.

Work performed in 2010:

1. Continuing the work from 2009, a program was created to select a database of pulses suitable to be studied regarding the discrepancy in electron temperature measurements and the identification of distortions in the Maxwellian distribution. The program is based on the fact that the discrepancy in temperature measurements between the Electron Cyclotron Emission and Thomson Scattering diagnostics appears in the ECE spectra as a difference between the temperature deduced from the second and third (optically thick) harmonic, the third harmonic being lower than expected from Maxwellian simulations and agreeing with the Thomson Scattering electron temperature. The study of the selected pulses has begun as soon as the database was checked and finalised. The pulses are studied carefully in detail in order to find cases of pulses with similar plasma conditions that could be compared directly, to identify any correlation of the discrepancy with any plasma parameter and to confirm or reject previous assumptions associated with the discrepancy. In conclusion, it seems that the disagreement between the ECE and Thomson Scattering diagnostics is based on the Maxwellian assumption (and whether it is valid or not in plasmas with strong auxiliary heating). However, further investigation is needed, which will not involve the latter assumption.

2.3 Plasma fuelling

2.3.1. 3-D pellet modelling (fuelling, drift, ignition with pellets)

Background and Objectives: The 3D pellet modelling activity is performed in collaboration with IPP Garching, and involves the development of multi-dimensional resistive MHD codes for pellet clouds and pellet ablation. The long-term objectives of this activity are to develop multi-dimensional pellet codes, and possibly a 3D resistive MHD pellet code, for pellet-plasma interaction studies, pellet fuelling of magnetic fusion devices, and ignition of magnetically confined plasma with pellets. So far a 2D+1 MHD code has been developed, which computes the ablation rate self-consistently. A number of scenarios have been performed with the 2D+1 code; results are yet to be analysed. In parallel a 3D MHD pellet code is being developed, this 3D code has no need for an input parameter z₀ which is necessary for the 2D+1 code.

Work performed in 2010 (in co-operation with IPP-Garching):

1. A number of scenarios have been performed with the multi-cell version of the 2D+1 MHD code, with the ablation rate computed self-consistently. Specifically, computational results on the ablation rate have been obtained for a pellet with initial radius of 5 mm and varying the pellet injection velocity from 125 m/s to 3000 m/s, for magnetic field strength of 1.5, 3.0, and 6.0 Tesla. Forty five scenarios, where the ablation rate is computed self-consistently and the pellet has been totally ablated, have also been performed. These scenarios involve 3 different pellet size (1, 2, and 3mm pellet radius), 3 different pellet velocities, and 3 different magnetic field strengths. Performing these scenarios was very cpu time intensive, where tens of thousands hours of cpu time have been used. The results from these scenarios will be analysed and if possible compare them with results from the 3D MHD code when available.
2. With the work on the 3D resistive MHD code a Poisson type algorithm has been implemented for enforcing divB = 0 numerically. This algorithm is very cpu intensive, and the constrained transport algorithm will be examined in the near future. Numerical experiments have been performed on the expansion of a high density blob, in 3D Cartesian geometry, with initial magnetic field to be parallel to the z-axis (or any other axis, x or y). The 3D resistive MHD code has been modified and moving pellet (or stationary) ablating self-consistently has been implemented (see Annex 24) and some preliminary results have been obtained. The code is under development and testing.

2.4 Real Time Measurement and Control

2.4.1. Automatic control of MHD instabilities

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

Work performed in 2010:

1. In this period, the construction of a closed-loop control algorithm for ECCD-based NTM stabilisation was initiated. A block system design for the feedback controlled process, the ECCD-driven NTM dynamical evolution including perturbations (i.e. edge density fluctuations), diagnostic sensor response and actuator control, was defined, and the numerical simulation is carried out in Simulink. As a first step, the block for the open-loop system response, based on a MATLAB routine for the solution of the modified Rutherford equation, was completed and benchmarked against a Fortran routine solving this specific equation [see 1.3.1(ii) and Annex 3]. Then, we have initiated the development of the blocks for the diagnostic sensors, based on the physics equations governing the measuring processes of the magnetic island parameters (island width and diamagnetic rotation), and for the ECCD actuator, using an estimation of the proper initial wave launching conditions based on the beam tracing computation of the backward propagation from the desired deposition point towards the plasma edge.