B2 Development of Plasma Auxiliary Systems

2.1 Heating and current drive systems

2.1-1. Application of sheet e-beam to quasi-optical gyrotrons

Background and Objectives: Gyrotron oscillators employ a weakly relativistic electron beam to produce coherent radiation of approximately 2 MW in the range of 170 GHz, required for electron cyclotron heating in fusion applications like ITER. 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. This configuration results in the amplification of the initial Gaussian-shaped output of the gyrotrons to the power levels of several MW that are needed for an efficient fusion reactor.

Work performed in year 2007 (in co-operation with CRPP and FZK):

1. The self-consistent iterative process required to monitor the interaction evolution has been implemented successfully both in Matlab and C++. Extended self-consistent simulations took place in order to compare both versions of the interaction codes, and in all cases, the comparison has been found very satisfactory. The obtained results strongly indicate that the initial output beam of a co-axial gyrotron of approximately 2 MW can be effectively amplified to the range of 10 MW or more, depending mostly on the power carried out by the electron beam. Details are presented in Annex XVII.
2. The above single-processor codes are under continuous improvement in order to overcome numerical inconsistencies and speed up the calculations; nevertheless they are relatively slow and fail to generate massive results that would allow a thorough study of the interaction. Therefore, substantial effort has been invested in creating a parallel C++ code, in particular, to apply the Message Passing Interface (MPI) for the management of data transfer between multiple processors in parallel computer architecture. The parallel version of the code works properly, in agreement with the results obtained by the single-processor version, and gives the opportunity to solve large demanding simulations as it exploits a greater amount of CPU memory resources.
3. Extended testing and benchmarking of the parallel code is carried out in order to determine the optimal parameters for fast as well as efficient and reliable simulation results that will also make clear the potential and the limitations of the code. (Work to be continued into next period.)

2.1-2. Electromagnetic code for beam-tunnel spectrum and slotted coaxial gyrotron 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 the domain of validity of the aforementioned model.

Work performed in year 2007 (in co-operation with CRPP and TEKES-UofLatvia, but also of interest to FZK):

1. Beam loading and its effects in a gyrotron beam tunnel (continued from previous period): Numerical results have been produced to determine the effect of the losses on the development of all kind of modes (TM, TE and hybrid) due to the beam-wave interaction in a gyrotron beam tunnel. From preliminary results for geometries found in the literature several discrepancies have been revealed. After an extensive investigation, several normalisation mistakes have been revealed and corrected. Subsequently, the numerical results obtained by our codes have been in excellent agreement with those presented in the literature for backward-wave oscillators operating mainly with TM modes (see Annex XVIII). Furthermore, special effort has been given to simulate the gyrotron beam tunnel for the W7-X experiment of Wendelstein. This work is of great interest for FZK, since several damages have occurred at the beam tunnels of the corresponding gyrotrons due to the parasitic oscillations (mainly of TE and hybrid modes), which have been observed experimentally. Finally, the mathematical formulation of TM modes for a metallic waveguide with any arbitrary periodic surface corrugation has been done, the corresponding numerical code has been developed and test runs have been performed in part. (This work will be finalised in the first months of 2008.)
2. The mathematical formulation, previously obtained for the study of TE modes in the resonator of the coaxial gyrotron, has been extended to include TM modes. Based on this formulation, a numerical code has been developed to calculate the eigenvalues, the electromagnetic field profile as well as the ohmic losses of the supported TM modes (see Annex XIX). In parallel to this, the TE version of the code has been used to examine the continuous frequency tuning possibilities of coaxial gyrotron cavities working in various frequencies. Several linear and nonlinear tapering profiles have been tested with promising results (see Annex XX).
3. Beam loading in a coaxial waveguide with circumferential corrugations (continued from previous period): Several subroutines have been developed and a few numerical tests have been performed for the cases of TE and TM modes. This subtask has been delayed since G. Anastasiou, who was the principal researcher of this subtask, has withdrawn (for personal reasons) in 2007.

2.1-3. Coaxial and harmonic gyrotrons

Background and Objectives: This activity addresses the coaxial gyrotrons, which have been seen as the most promising configurations of high-power, high-frequency RF sources for ECRH heating. It is performed in close interaction with the European gyrotron development programme and has to be adjusted to the evolving needs. Therefore, in 2007, it has also been extended to conventional gyrotrons, in view of the design of a 170 GHz, 1 MW conventional gyrotron (initiated as an EFDA task on July 2007), which is the EU back-up gyrotron for ITER. 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 2007 (in co-operation with FZK and CRPP):

1. The simulations performed by the NTUA interaction code, regarding the pre-prototype 170 GHz coaxial gyrotron existing at FZK, indicated possible excitation of parasitic modes. The presence of those modes, which are the radial satellites of the operating mode, may explain the observed discrepancy between the experimental results of the FZK pre-prototype and the simulations by several interaction codes. The explanation of this discrepancy is of top priority, therefore numerical as well as theoretical studies on the excitation of the radial satellites have been carried out. Details can be found in Annex XXI. However, further comparison with experimental results was not possible, because, in 2007 there was no rf output from the 170 GHz prototype coaxial gyrotron at CRPP.
2. The subroutine of the NTUA interaction code, relevant to the calculation of the electron motion, has been modified in order to use a numerically more efficient mathematical model (involving a complex differential equation instead of a pair of real differential equations). By this modification, the interaction code became three times faster. A subroutine for calculating the axial field profile self-consistently in the case of steady-state, single-mode operation has been developed. Further optimisation and verification of this subroutine, together with extension of the self-consistency to the time-dependent, multi-mode part of the code are planned for the next period.
3. Since the experimental performance of the pre-prototype 170 GHz coaxial gyrotron at FZK is not ideal up to now, an investigation for alternative configurations of its coaxial cavity has been performed, beginning with alternative coaxial inserts. The results are presented in Annex XXII.
4. An appropriate interface between the NTUA interaction code and the ARIADNE beam tunnel code has been prepared, in order to study the effect of azimuthal inhomogeneity of electron emission on mode competition and gyrotron performance, considering realistic electron beams. However, pertinent simulations have not been performed yet, because of the large amount of effort dedicated to the high-priority tasks (i) and (v) (see below).
5. All NTUA codes have been extended to apply also to conventional gyrotrons, in view of the design of the 170 GHz, 1 MW EU back-up gyrotron for ITER. Operating-mode selection for this gyrotron has been carried out, along with preliminary cavity designs for the most promising modes. (Work performed as a voluntary contribution, adjusted to the needs of an EFDA task, to be continued into next period.)

2.1-4. Chaotic electron dynamics in gyrotron resonators

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 2007 (in co-operation with TEKES-UofLatvia):

1. The self-consistent model for electron interactions with rf modes has been defined and the methodology of the approach for studying coupled electron – wave dynamics has been determined. The method consists of the utilisation of the analytical results obtained in our previous studies for fixed-field electron dynamics along with an integral representation of the solution of the wave field equation, in an iterative fashion. Preliminary investigations of the feasibility of implementation of a numerical code for the aforementioned iterative procedure have been performed. Further a-priori investigation on the purpose of implementing such a code has to be performed in comparison to other approaches and codes, already used by the respective community. (To be continued into next period.)
2. Preliminary comparisons have been performed between the numerical results and the analytical ones (obtained in our previous studies), regarding the electron distribution function as well as calculations of collective electron characteristics, for cases of moderate values of beam to rf coupling.
3. A near-symplectic explicit integration scheme has been obtained for the fast calculation of electron trajectories in gyrotron cavities. This integration scheme has several advantages in comparison with standard schemes (e.g. Runge-Kutta methods) as well as implicit symplectic schemes. This study is quite general and applies not only to electron dynamics in gyrotron cavities but also to a large variety of Hamiltonian systems of interest to fusion research, such as ECRH and others. (For more details see Sec. 1.5-1 (iv) and Annex XIV.).

2.1-5. Novel configurations for a gyrotron collector

Background and Objectives: In a gyrotron, the power of an electron beam (of electrostatic origin) is converted to electromagnetic, at a conversion efficiency ηe (typically, about 30%). Subsequently, by employing a collector with depressed voltage, part of the power in the used electron beam can be converted back to electrostatic, with efficiency ηcoll. The resulting overall efficiency is equal to ηtot = ηe/[1 –ηcoll(1–ηe)]. This expression brings out the importance of an efficient collector. In conventional depressed collectors, the efficiency is limited by the energy spread in the beam electrons (which is generated by the interaction in the cavity and therefore unavoidable in a high-power gyrotron). However, preliminary considerations have been recently given (by Dr. J. Pagonakis at CRPP) on a novel approach, in which suitable electric fields are meant to be used to segregate the electrons according to their energy after the cavity (thus avoiding the main draw-back in the design of an efficient electron beam collector), by taking advantage of the ExB drift.

Work performed in 2007 (in co-operation with CRPP):

1. Our contribution consisted of:

• Assessing the concept: The concept appears to be solid in its conception. For the (simplest) idealised case of a filamentary beam, with arbitrary energy content, but exclusively in the axial direction, expressions have been obtained for the appropriate shape of the collector boundary and the (continuous) distribution of the electrostatic potential along it, for the efficiency of the collector to be equal to 100% (with the added benefit, of zero thermal loading of the collector walls).
• Considering alternative geometries (in co-operation with Dr. Pagonakis) to implement the concept, for an electron beam like the one typically employed in a gyrotron (i.e., one with annular cross section). As most promising has been identified one with a coaxial collector (with the inner and outer parts each collecting half of the beam).
• Performing analytical calculations for the effects on the collector efficiency by features like beam thickness, finite (but small) transverse velocity, finite Larmor radius and step-wise distribution of the electrostatic potential along the collector surface. It has been found that these features reduce somewhat the efficiency, but for realistic values of the parameters this reduction appears to be small and the efficiency remains around 90 %). (See Annex XXIII for the details of the calculations.)
• Studying the results of numerical simulations (performed by Dr. Pagonakis) and comparing them to the theoretical expectations.

2.2 Plasma diagnostics

2.2-2. Calculation of electromagnetic field distribution on TORPEX

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, a search for optimal transition configurations, which minimise the reflection coefficient at the excitation port, will be performed. For all these calculations the commercial code MAFIA will be used.

Work performed in 2007 (in co-operation with CRPP):

1. Calculate full spectrum of waves excited: Several more realistic geometries regarding the real TORPEX structure have been examined. Specifically, the electromagnetic field distribution of cylindrical waveguides with ohmic losses excited by a rectangular port and an appropriate transition have been calculated, as well as the electromagnetic field distribution of a toroidal segment with ohmic losses excited by rectangular port and an appropriate transition. For these simplified models it has been seen that the electromagnetic field is not homogeneous in the cross-section; it is mainly concentrated into the centre of the structure (when the transition is present) and only a few modes propagate (although a huge number of modes are excited). For this reason, after discussions with the members of CRPP, it has been decided to study a cylindrical waveguide with a dielectric slice with high relative permittivity (~ 100) and losses. This case is a simplified model of the real structure with the plasma inside. From the numerical simulations it has been seen that the electric field distribution at several axial positions inside the waveguide seems to consist of several modes, it is not homogeneous and is concentrated in the area before the dielectric material, which acts as a reflector due to its high values of relative permittivity (see Annex XXIV).
2. The influence of the existing transition at the excitation port on the field distribution in the case of a cylindrical waveguide as well as for a section of a toroidal waveguide has been examined. From the numerical results it has been seen that the presence of the transition, although it improves the concentration of the field in the centre of the structure, increases significantly the return losses at the excitation port. Nevertheless, since the real structure contains plasma (inhomogeneous dispersive dielectric material), optimised transitions will be considered for more realistic models of the actual structure.

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 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. The multi-dimensional code development involves: a 2-D+1 pellet-plasma code (2-D Eulerian resistive MHD for the poloidal plane and Lagrangian for the 3rd direction.), a 2-D resistive MHD code in Cartesian geometry for the equatorial plane, a 2-D cylindrical resistive MHD code, and the implementation of pellet modules in a 3-D MHD code, the M3D code developed by Princeton University.

Work performed in year 2007 (in co-operation with IPP):

1. A number of runs, with varying anomalous (perpendicular to the magnetic field) thermal conductivity, were performed with the 2-D+1 code. The purpose of varying the anomalous thermal conductivity was to reproduce the experimentally observed radiation patterns (see Annex XXV). A two-dimensional interpolating routine has been developed for interfacing the output from the equilibrium code DIVA to the 2-D+1 code. Three-component magnetic field has been incorporated in the 2-D+1 code. Also, a “Poisson type” routine to enforce div(B) = 0 (for the two components of the magnetic field in the poloidal plane) has been developed. This module has been tested only for an initial uniform magnetic field.
2. The objective of this task is to have a more accurate description of the expansion of the code in the 3rd direction and hence to couple an ablation code (a code which computes the ablation rate self-consistently) to the 2-D+1 code. The 2-D Eulerian code for the equatorial plane has been coupled to the 2-D+1 code. Only some specific modules of the 2-D equatorial code have been activated (temperature diffusion equation and flow in the z-direction), as comparisons with multi-cell Lagrangian modules must be made first. The cylindrical 2-D resistive MHD code is still being developed, while its fluid part has been tested. Also a 1.5-D  MHD code has been developed (in co-operation with TUC).