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: The development of the gyrotron oscillators as the dominant and most effective high-power microwave source is motivated by the high-power demands required for plasma heating and current drive in fusion tokamaks. Gyrotrons employ a weakly relativistic electron beam propagating along a strong magnetostatic field in a microwave resonator. 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 with the rf beam produced by the gyrotron. 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 successful and efficient fusion reaction.

Work performed in year 2009 (in co-operation with CRPP and KIT):

1. The power-flow generated by the radiation field was calculated on the surface of the orthogonal interaction area in order to verify the previously obtained results that implied a substantial power gain without any significant off-axis power distribution (see Annex 14).
   Based on the numerical results for the electron beam trajectories representing a space current density, we calculated the radiation components accordingly, and the first steps towards an analytical expression for the EM field distribution have been made (see Annex 15).
2. Preliminary calculations using a fully-relativistic approach indicated that the weakly-relativistic approximation of the electron beam works adequately in the range of parameters of our simulations. Therefore the work in the fully-relativistic approach has been considered as of lower priority.
3. Parallelisation of the respective code has implemented and is under continuous optimisation. Simulations of large number of cases have thus been made possible and are used in order to determine the optimum set of parameters for an efficient and numerically consistent operation.

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. In addition, harmonic interactions are also studied, for the purpose of producing high frequency at reduced magnetic field requirements. This activity 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 advances 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 and to studying the fundamentals of the gyrotron interaction. 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 2009 (in co-operation with KIT, CRPP and UofLatvia):

1. The advanced simulation concept of using a non-constant frequency as the carrier frequency of a mode was investigated. Validating comparisons between the NTUA self-consistent interaction code in the package EURIDICE and the corresponding code SELFT (existing at KIT) were performed for multi-mode cases. We note that from March 2009 this subtask has been included in the F4E grant agreement F4E-2008-GRT-08 and therefore its man-power has been excluded from CoA. Details on the continuation of the work can be found in the reports associated with this grant.
2. This subtask has been included in the F4E grant agreement F4E-2008-GRT-08 and therefore its man-power is excluded from CoA. Details on the performed work can be found in the reports associated with this grant.
3. A slow-time-scale, single-mode model for the equations of electron motion in an axially varying magnetostatic field and in the presence of TE, TM or hybrid modes (close to or far from cut-off) was derived. The aim is to study the parasitic oscillations in the gyrotron beam tunnel far from the small-signal regime. The model takes into account the axial non-uniformity of the magnetostatic field even for relatively high field gradient. Moreover, a correction to account for the axial variation of the beam tunnel average wall radius has been introduced by expressing the fields by the WKB approximation. A numerical implementation of the derived model, in the case of waveguide modes, was carried out by developing a completely new code. The new code is at the stage of verification. Optimisation of the code and linking to other codes for a more complete field representation are foreseen, as being necessary for obtaining useful results. At the same time, the influence of an axially varying magnetostatic field on the electron parallel and transverse velocity was mathematically formulated in a way suitable for extending the existing self-consistent interaction code in EURIDICE to incorporate this effect. After the relevant numerical implementation (foreseen for the next period), the existing interaction code will be able to take into account more accurately an axially varying magnetostatic field. Finally, approximate results on parasitic beam-tunnel oscillations in existing gyrotrons were obtained by the existing interaction routines in EURIDICE after appropriate modifications. This was done within the F4E grant agreement F4E-2008-GRT-08 and therefore the related man-power has been excluded from CoA. Details can be found in the reports associated with this grant.
4. Regarding code improvement, significant advances in computation speed were achieved and extensions incorporating more sophisticated modelling, involving RF space-charge and varying axial electron velocity, were implemented. More detailed information on those improvements can be found in Annex 16. Regarding code verification, comparisons with the self-consistent codes SELFT (KIT) and COAXIAL (UofLatvia) were performed for multi-mode cases with satisfactory results. More significantly, however, the comparisons with the interaction routine in the code Ariadne++ (existing at CRPP), initiated in the previous period, were continued. These comparisons aim to asses the amount of error introduced on the electron motion by the usual approximations used for fast simulations in EURIDICE (but also in SELFT and COAXIAL). The routine in Ariadne++ uses far less approximations for the electron motion in a given field. Details on the results can be found in Annex 17. Due to the extended effort devoted to the aforementioned improvements and verification, the work on a generalised surface impedance model for the ohmic losses will be initiated in 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 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 2009 (in co-operation with CRPP, KIT, UofLatvia and PSFC/MIT):

1. To check the validity of the results obtained extensive numerical tests have been performed for several different as well as limiting cases of gyrotron (conventional and coaxial) beam tunnels. From these tests, several minor errors have been revealed and corrected in the codes. It has been also seen that the results for TE and TM modes are identical with those of hybrid modes with m = 0, as it is expected. Moreover, both codes BeamFishbone (for conventional gyrotron beam tunnel) and CoaxBT (for coaxial one) have been parallelised and checked. Therefore they are now ready to be run in parallel machines available in Greece (UoAthens and NTUA), Switzerland (CRPP) and Germany (KIT). Τhe case of non-periodic surface corrugation profiles has been included in the numerical codes for all kind of modes and checked (see Annex 18). Furthermore, the mathematical formulation for the case of an inhomogeneous magnetostatic field has been initiated and several analytical expressions have been properly addressed. However, the formulation is not yet finalised and is expected to be ready in the following year.
2. In the previous years we have developed the mathematical formulations and the corresponding numerical codes to study the dispersion properties of TM and TE modes in a circular waveguide with surface corrugation, which is described by a periodic function R(z). In 2009, the above formulations have been extended to include the beam loading in the small-signal regime for both types of modes (see Annex 19). In particular, the analysis is based on the linearised Vlasov equation in order to investigate any possible parasitic oscillation that may be excited from the noise and has been presented in detail in Annex 24 of 2008 Annual Report. This updated formulation has been included in the code for TM and TE modes and testing procedure has been initiated.
3. Note that this subtask is included in the F4E grant (F4E-2008-GRT-08) and therefore its man power is excluded from CoA.
4. For this subtask there is no substantial progress, because the of restricted personnel availability due to more urgent subtasks. Nevertheless, several initial discussions have been done during the mobility missions at CRPP and KIT in the beginning of 2009.
5. This subtask is related to the previous task (iii) and it is included in the F4E grant (F4E-2008-GRT-08); therefore its man power is excluded from CoA.

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 2009 (in co-operation with UofLatvia):

1. Feasibility studies on the application of Hamiltonian perturbation methods on the self-consistent system describing both electron motion and rf mode dynamics have been initiated. The estimation of the expected results and advantages of this approach in an ongoing task.
2. A Hamiltonian system describing both electron motion and rf mode dynamics has been derived. The system allows for a unified formulation of gyrotron cavity dynamics in a self-consistent fashion. In comparison to commonly used iterative approaches where electron motion equations and rf mode wave equations are solved in different steps, the present formulation is expected to have several advantages, with respect to analytical and numerical studies. Considerations on the numerical implementation of this approach and its potential advantage in comparison to existing approaches, already incorporated in existing codes, have been initiated.

2.2 Plasma diagnostics

2.2.1 Numerical simulations for fusion electrodynamic systems

Work performed in 2009 (in co-operation with CRPP):

For this activity no substantial work has been done in 2009, since we have concentrated our efforts in other more urgent tasks of CoA [mainly 2.1.3(i), and (ii)].

2.2.2 Diagnostic methods for the measurement of electron temperature

 The systematic disagreement observed in electron temperature measurements between the Electron Cyclotron Emission and Thomson Scattering diagnostics in high temperature plasmas it is a long-standing problem of tantamount importance from the technological point of view and the physical understanding as well. Since suitable human resources became available within 2009, this issue was incorporated in the priorities of the Association.

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 2009 (in co-operation with JET):

1. The Electron Cyclotron Emission spectrum was studied extensively concerning the appearance of the discrepancy in temperature measurements between the Electron Cyclotron Emission and Thomson Scattering diagnostics in high temperature plasmas. The study of the spectra aimed at the exclusion of possible causes for the discrepancy, such as calibration problems of the ECE diagnostic and known errors of both diagnostics, as well as the identification of the conditions under which the discrepancy appears. In this direction, measured spectra were compared with simulations and a programme was created in IDL so as to generate a database of suitable shots to be studied, simulated and compared. More details can be found in Annex 20.

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 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, (b) radiation patterns produced by ablatant clouds and particularly investigation of visible striations aligned with B_|| direction, and grad(B)-induced drift phenomena, and (c) possibly apply this code for studying the injection of hydrogen pellets into helium plasmas, since ITER before going nuclear will be operated with helium plasma and hydrogen pellet injection. So far the objective of computing the ablation rate of moving deuterium pellets in hot plasmas self-consistently has been achieved; of course computing various pellet injection scenarios and comparison with experimental data are necessary.

Work performed in 2009 (in co-operation with IPP-Garching):

1. The 2-D+1 code with the single Lagrangian cell approximation in the z-direction (Plasma Phys. Control. Fusion 50 08500 (2008))has been modified into a multi-cell approximation. In the 2-D+1 code (2-D Eulerian for the poloidal plane and 1-D Lagrangian along field lines, z-direction) to each mesh point of the Eulerian grid, a toroidal flux tube is attached. The field-aligned expansion (heating etc.) of the ablated substance is confined to these flux tubes. The corresponding changes of the state parameters are taken into account in a Lagrangian approximation. Uniform Lagrangian cells are generated in the flux tubes by splitting the first Lagrangian cell at the symmetry plane. The criterion for cell splitting is the approximate constancy of the cell mass.
   The ablation rate, which is the source term of the rhs of the mass continuity equation, has been previously implemented as an assigned parameter or computed from scaling laws. Now the ablation rate of moving pellets is computed self-consistently by considering the energy fluxes affecting the pellet surface thus causing pellet erosion (see Annex 21).
   Because of lack of manpower parallesation of the multi-cell 2-D+1 code has not been made. Work on developing a truly 3-D Eulerian resistive MHD code, in Cartesian and cylindrical geometries, has been initiated. The routines pertaining to the hydrodynamics in Cartesian geometry have been developed and tested.
2. To the 2-D axisymmetric resistive MHD code an ablation module, similar to the 2-D+1 code, has been implemented. This is relevant to stationary pellet. Initial magnetic topologies, computed analytically, for open mirror systems have been implemented in this cylindrical code. A large number of runs, with high density high temperature plasmas and initial open system magnetic topologies, have been performed (see Annex 22).