2.1 Heating and current drive systems

In 2011, following the response of the Association to the Call for Participation to the Task Agreement WP11-DAS-HCD-EC (DEMO Design Assessment Studies, HCD&FP, Electron Cyclotron), issued by PPP&T, a series of assessment studies was undertaken. They focused on the present status and on the roadmap for further development, in order to meet the DEMO needs, of (a) the modelling and numerical tools used for gyrotron studies, and (b) the innovative concepts for improving gyrotron power and efficiency. A contribution to a common report, prepared by all the Associations involved, has been delivered. More details on the Association’s assessment studies can be found in Annex 14, which is cited in the common report.

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 2011 (in co-operation with KIT, CRPP, and UofLatvia):

(i) EURIDICE was improved, regarding energy conservation and stability, through the implementation of the concepts of time- and space-varying reference frequency, as well as the use of an advanced method for frequency calculation. Moreover, EURIDICE was extended to consider the effect of the transverse high-frequency magnetic field BRF⊥ on the interaction. The interface between EURIDICE and the electrostatic 3-D code-package ARIADNE, existing at KIT, has been upgraded by incorporating the concepts of macroelectrons and of non phase-mixed beams, in order to allow improved studies of the influence of electron-beam azimuthal inhomogeneity and/or misalignment on beam-wave interaction. More details on the above extensions and improvements can be found in Annex 15. Some initial verification comparisons with TWANG (CRPP), regarding the BRF⊥ influence, were performed. Verification comparisons with SELFT (KIT), relevant to the influence of the magnetostatic field on electron momentum, were also made. In addition, the investigations on the experimentally observed ~130 GHz parasitic oscillations in the 140 GHz, 1 MW gyrotron for W7-X, currently believed to be due to after-cavity interaction (ACI), have been continued; yet with no conclusive results. The numerical implementation of the advanced multi-frequency integral boundary condition in EURIDICE was attempted, following the mathematical formulation obtained in the previous period. However, serious stability problems were encountered. The reasons are under investigation. Further work on the mathematical formulation of the integral boundary condition and its numerical implementation is foreseen for the next period.

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

Background and Objectives: 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 negative 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. The activity also aims at even more realistic modelling of electromagnetic waves in dielectric loaded beam tunnels and at providing basic design directions towards efficient suppression of parasitic oscillations. The methods used for the more realistic modelling involve application of open boundary conditions by means of Fourier transform techniques and semi-analytic treatment of the problem. In parallel, slotted coaxial gyrotrons cavities, used to facilitate mode selection, are addressed. Such structures are typically calculated by employing the model of distributed impedance (SIM) 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 the domain of validity of the aforementioned model.

Work performed in year 2011 (in co-operation with IHM/KIT, CRPP/EPFL, PSFC/MIT):

(i) Numerical results have been produced for several beam-tunnel geometries provided by KIT and CRPP. These results have been compared with those obtained by codes available at both institutes as well as with the commercial tool CST Studio Suite. Furthermore, several optimisations were introduced in our in-house codes. In addition, parametric studies were performed to understand the effect of the geometry and dielectric material on the parasitic modes in a stacked gyrotron beam tunnel (Annex 16). Regarding coaxial cavity modelling, several investigations were performed aiming at understanding the way that the spatial harmonics contribute to the reformation of the eigenvalue spectrum and to the distribution of the ohmic loading on the wall. It was realised that although the full-wave model gives approximately the same results for the eigenvalues, this is not the case for the ohmic loading. Based on these observations it was attempted to create a simple extension in the SIM losses calculation procedure in order to benefit from the full-wave model without implementing it in the SIM codes (Annex 17). Due to the lack of man-power resources the numerical studies for other types of corrugations were not performed in 2011.

(ii) We initiated the development of a new fully numerical code based on the FDTD method to solve the Maxwell equations in wave guiding structures. This activity aims, in the long term, at developing 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. The code developed during 2011 can currently handle only orthogonal geometries (Annex 18, Annex 19).

(iii) The electromagnetic model and code developed in 2010 has been extensively tested and considerable physical insight on the problem has been gained due to the analytic approach. Some possible design principles have been recognised but their final validation should be tested with accurate realistic simulation tools. For this reason we enhanced the electromagnetic model with the accurate transverse boundary conditions of the cylindrical geometry. This consideration does not require any approximations and our new improved model is accurate and realistic (see Annex 20). As a consequence the analytic character of the solution has been lost. This is not a problem as the new method remains numerically efficient involving rapidly convergent series and requires only a matrix inversion. We modelled also the whole structure with many unequal rings as well as the recess region of vacuum that protects the dielectrics from escaped electrons. As the improved model considers the realistic boundary conditions it can be further enhanced in the future to be a reliable tool for realistic beam tunnel simulations. The consideration of more general longitudinal boundary conditions (e.g. imposing arbitrary reflection coefficients) was postponed and the available manpower was directed to the implementation of the new accurate model in a computer code. The new code was developed from scratch, was tested and the first preliminary results were extracted.

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 order to overcome the current physical and technological limitations that conventional high-power sources like gyrotrons exhibit, especially in view of the extensive power demands of the fusion reactors beyond ITER, a new conceptual configuration is proposed with the aim of generating high power and coherent RF radiation for efficient ECRH/ECCD. This concept involves an adequately powerful (nonetheless in a low current density) sheet electron-beam propagating along a magnetostatic field that interacts with the Gaussian RF beam provided by a gyrotron. The induced radiation field that the wave- particle interaction generates, is shown to be constructively added to the initial Gaussian field, providing a greatly enhanced amount of power.

Work performed in 2011 (in cooperation with CRPP and KIT):

(i) The calculations regarding the self-field radiation are properly expanded in order to incorporate the fully relativistic expression of the fields, with the aim of obtaining, accordingly, a more precise representation of the wave-particle interaction. The obtained expression constitutes of a periodic function of the wave-particle phase difference, and has a Fourier series representation (see Annex 21). However, higher-order trigonometric functions involved in these calculations obstruct the progress of this task, as multi-valued expression may occur. Therefore, the origin of the power inconsistency observed in our simulations and which could probably give rise to additional radiation effects, like the emergence of recoil-radiation, cannot be safely and precisely determined.

(ii) The parallel instance of the computer code that also includes the results of the previous task is not yet implemented because of the intrinsic numerical complications (see Annex 21) encountered. Its conclusion will be reassessed for the following year, in relation to the scheduled priorities.

(iii) Additional techniques including analytical expressions (see Annex 22) for the current were carried out in order to incorporate a more precise approximation of the radiation components. In particular, considering a suitable expression of the current that represents the electronic motion of the gyrating ensemble by dropping the individual kinematic parameter of each electron and by adopting mean-value approximations, we introduced an additional radiation component, namely the dipole-radiation component of the current-loop that represents the rotating bunch. The contribution of the single-component radiation (as only the component parallel to the magnetostatic field survives) (see Annex 21) to the initial field was directly built into our (semi-relativistic) calculations, and the corresponding simulations revealed that such an additional component is of minor significance for the range of parameters taken under consideration, and can be currently ignored, at least until task (i) is fully resolved.

(iv) The range of parameters leading to an efficient and successful interaction is currently determined to a great extent. However, it is evident that it strongly depends on the kinematic characteristics of the electron beam and the wave characteristics of the RF beam, as well as on the numerical implementation of the interaction scheme, and especially on the way that these factors coexist and govern the interaction. This fact implies that the outcome of each simulation cannot be estimated based only upon the applied initial conditions, but instead (see Annex 21), each case needs to be addressed independently, by optimizing accordingly (either analytically or numerically) the interaction parameters.

2.3 Plasma fuelling

2.3.1.3-D pellet modelling (fuelling, drift, ignition with pellets)

Background and Objectives: The 3-D pellet modelling activity is performed in collaboration with IPP Garching, and involves the development of a full 3-dimensional resistive MHD code for pellet clouds and pellet ablation. The long-term objectives of this activity are to develop multi-dimensional pellet codes, for pellet-plasma interaction studies, pellet fuelling of magnetic fusion devices, and ignition of magnetically confined plasma with pellets. A 2-D+1 pellet code (2-D resistive MHD for the poloidal plane coupled with Lagrangian modules for expansion along magnetic field lines) has been developed which computes the ablation rate self- consistently. The effort is now concentrated in the development of the 3-D resistive MHD code. Emphasis is given on the self-consistent calculations of the ablation rate because of differences in scaling laws (from experimental data) of the dependence of the pellet penetration depth on the magnetic field strength.

Work performed in 2011 (in co-operation with IPP-Garching):

(i) A number of scenarios have been performed with the 2-D+1 code in order to obtain scaling laws for the penetration depth of injected pellets (ablation rate). However these calculations are quite sensitive to the input parameter z₀, and more emphasis is now being placed in the development of the full 3-D pellet code. A number of scenarios have been performed with the 3-D code, which were also performed by the 2-D+1 code, in order to see the dependence of the magnetic field on the ablation rate. For stationary pellets it has been seen, in both codes, that the ablation rate increases with increasing magnetic field strength. It is noted that the self-consistent calculation of the ablation rate is presently based only on the thermal fluxes irradiating the pellet.

(ii) Scenarios with initial magnetic field, in the z-direction, decreasing with respect to the x- direction have been performed. Numerical modules have been developed for interfacing structured or unstructured grids (from equilibrium codes) to the grid of the 3-D code (and the 2-D+1 code). An ITER equilibrium from the HELENA equilibrium code has been given and interfaced as initial plasma profile and magnetic topology for the 2-D+1 (and 3- D) code. Scenario calculation with the 3-D code with moving and stationary pellets will be performed when the dependence of the ablation rate on the magnetic field strength has been clarified

(iii) The 3-D resistive MHD Cartesian pellet code has been tested for different number of planes in the z-direction. The descritasation in the z-direction is non-uniform and models up to the half of the torus. The self consistent ablation rate has been implemented, and for stationary pellets comparison with the 2-D+1 code has been performed. For moving pellets with high injection velocities (> 500 m/s) noticeable oscillations appear in the self- consistent ablation rate. These oscillations are grid dependent; they depend on how the ablated particles are deposited onto the numerical grid (see Annex 23). The present calculations of the ablation rate are based only on thermal fluxes irradiating the pellet surface. Stopping length calculations, for heating the cloud around the pellet surface must also be considered.

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 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 is to benchmark the established system identification methods on the accurate prediction of the (known as modeled) system dynamics.

Work performed in 2011(in cooperation with TEIPIR):

(i) In this period, the construction of a simulation algorithm in Simulink/Scicoslab for the closed-loop control of NTMs using ECCD was continued. During the previous period, a block-system design for the controlled process including perturbations, diagnostic sensor response and actuator control was defined, and the block for the open-loop system, based on the solution of the modified Rutherford equation, was finalised. During the current period, we completed 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. In experimental practice, the requirements for the mirror dynamic response are usually set to an angular error of 1ο and an angular velocity response of 20o/s respectively. Our results show that the closed-loop model responds to the control actions with almost no steady-state error and within the bounds set by the angular velocity requirements. A comparison of the closed and open-loop systems yields that the response time is significantly reduced due to the effect of the PID controller (see Annex 24). The development of the block system for the diagnostic sensors, based on the physics of the measuring processes of the island width and diamagnetic rotation, as well as for the plasma wave propagation, in terms of the asymptotic computation of the wave propagation between the proper wave launching point and the desired deposition point, are in progress.