2a) Beam-wave interactions and high-power RF generation

2a1) Application of sheet e-beam to quasi-optical gyrotron

Background and Objectives: A possible application of a sheet beam (whose stability has been studied earlier) is to have it interact with the propagating Gaussian RF beam produced by a conventional gyrotron. A simple preliminary calculation has indicated that the parameters of the RF beam from, say, a 2 MW gyrotron are adequate to extract significant amounts of power from the e-beam, with no need for feed-back from the walls of any cavity. Thus, if this work proves promising, a first-stage conventional high-power gyrotron could provide the initial generation of the RF beam with mode selectivity and high efficiency and the quasi-optical gyrotron concept can be applied as a second stage to multiply the RF power to the levels needed for a fusion reactor. (This is a multi-annual activity, performed in cooperation with CRPP and FZK.)

Work performed in year 2004: The work of the previous periods was continued, on assessing the feasibility of using a sheet electron beam on the output radiation beam from a conventional gyrotron directly (that is, without employing a resonator). The work performed has included the following:

(i) Three-dimensional effects: The modulated electron beam, represented by beamlets of simulation electrons moving with their axis of rotation perpendicular to the radiation beam, may not radiate along the incoming Gaussian beam as initially assumed and part of their energy loss is possible to raise unwanted off-axis radiation. The expressions representing the E/M fields produced due to the electron motion were carried out for a single electron and the field components at the plane of the electron beam were extracted for the case of perpendicular incidence. A computer code was implemented in order to calculate the overall contribution of all the electrons to the field, and the radiation pattern and distribution were obtained (seeAnnex I). The profile and characteristics of the produced radiation fields is under investigation in order to find out the way it adds to the incoming Gaussian RF fields and to determine the feasibility of a substantial amplification.

(ii) CARM-like interaction for the electron beam propagating at an angle relative to the radiation beam: Only the case of the perpendicular incidence was investigated, as early calculations showed up that the interaction is more effective in this vicinity and there is no need at present for further calculations at a different direction of incidence.

(iii) Effects of axial non-uniformity on design of beam tunnel for a sheet beam. In year 2001, the relevant equations for the electrostatic potential Φ (in order for the beam tunnel boundary to preserve the shape of a sheet electron beam) have been formulated, but only partially solved (due to the withdrawal of the scientist in charge). This work has been continued during year 2004. The second order (proportional to ∂²B/∂z²) correction for Φ has been obtained analytically (while the first order one vanishes identically) and the corresponding equi-potential surfaces have been studied (see Annex II for more details).

 

2a2) Self-consistent 3-D electrostatic code for gyrotron beam tunnel

Background and Objectives: Available electrostatic codes (E-GUN and DAPHNE) for the electron gun and beam tunnel assembly assume azimuthal symmetry and hence they are two-dimensional. As such, they cannot be used to describe situations without azimuthal symmetry, whether they arise out of construction imperfections (e.g., non-uniform emission from cathode, deviations from perfect alignment, etc.) or from inherent necessity for non-symmetric construction (e.g., a sheet beam for the quasi-optical gyrotron). To cover this need, this activity aims to prepare a self-consistent electrostatic code in three dimensions and to use it in gyrotron beam tunnel studies. (This is a multi-annual activity, performed in cooperation with CRPP and FZK.)

Work performed in year 2004: The work of the previous periods was continued in two directions, first to obtain results for the 3D behaviour of actual gyrotron guns, and second to improve on the performance and versatility of the code ARIADNE. More specifically, during the period in subject, the following work was done:

(i) The deterioration of the electron beam quality, due to an eventual non-uniform emission from the cathode in the coaxial Magnetron-Injection Gun of the 170 GHz gyrotron of FZK, has been studied with the use of the code ARIADNE. The average shifts and the spreads of the beam parameters (radial position, transverse and parallel velocity) have been calculated numerically and comparisons have been made to the case of a uniform system. In general, significantly larger spreads have been observed, which indicates that the emission mechanism or the nonuniformity play a very important role. Details are presented in Annex III.

(ii) Regarding upgrading of the code:

• An algorithm of automatic generation of database files of the programme ARIADNE has been developed and incorporated in the code facilities, for cases when the code runs for the first time for a new user account.
• A new set of advanced commands for the introduction of the beam tunnel geometry in the programme database have been incorporated. With the use of this set of commands the user has the possibility to define the three-dimensional description of segment geometry through surfaces represented by mathematical functions of the azimuthal angle and the axial position.
• The online help files have been improved.

(iii) The ARIADNE code has been installed in the parallel computer system PLEIADES of CRPP and with the cooperation of CRPP personnel a large number of test runs have been produced.

2a3) 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 normalized field amplitude, while energy exchanged 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 employing the model of distributed impedance and therefore the calculations are limited to the domain of applicability of this model. Thus, this activity also aims at the development of numerical codes, to calculate the frequency spectrum of slotted coaxial cavities, to allow performing calculations not constrained by this model. (This is a multi-annual activity performed in cooperation with CRPP and HUT/TEKES.)

Work performed in year 2004: The work of the previous period was continued into the period in subject, on addressing several issues related to the development of electromagnetic modes in geometries representative of typical gyrotron beam tunnels or interaction cavities. In particular:

(i) Beam loading and its effects in a gyrotron beam tunnel (continued from previous period): The analytical calculations for the beam loading were finished last year. This year, the corresponding subroutines have been written and implemented in the numerical code called Beam-Fishbone for the TM and TE modes. From the test runs performed, an empirical formula was derived to estimate the location of the roots of the dispersion relation. Along with some improvements made to the root finding method, significant acceleration was achieved to the code. Several numerical tests have been done and results have been obtained for the dispersion relation as well as for the field distributions. More details are given in Annex IV. (This work is a collaboration with CRPP.)

(ii) Azimuthal Bloch modes in a coaxial gyrotron cavity with a slotted inner rod (continued from previous period): We have finished the analytical calculations leading to a linear eigenvalue problem (i.e., the formulation of the field description in the empty part and inside the polar corrugations of the cylindrical waveguide, the application of the boundary conditions and the derivation of the final set of equations to be solved), we have formulated the ohmic losses on the conducting walls and the corresponding numerical code. We have also made several test runs and the numerical results for special cases coincide with those appearing in the literature. More details are given inAnnex V. (This work is in collaboration with Prof. Olgierd Dumbrajs from the Laboratory of Advanced Energy Systems, Helsinki University of Technology, Finland.)

(iii) Boundary conditions for partial reflections at the ends of the system, as appropriate (to be continued into next period): For this subtask no action has been taken, since our effort has been focused on finalizing the previous two subtasks, which are of high priority, also to our collaborating partners.

 

2a4) Coaxial and harmonic gyrotrons

Background and Objectives: Coaxial gyrotrons have the important advantage, that they provide more flexibility in suppressing the unwanted competing modes, in spite of the large-size cavities they employ, allowing for record-high power output. In addition, harmonic gyrotrons can produce extremely high frequency output, albeit at much lower power levels. For these reasons, coaxial and harmonic gyrotrons have been designed and studied. The numerical codes that have been used for this activity can be used for additional designs, while they also admit significant improvements. (This is a multi-annual activity, performed in cooperation with FZK.)

Work performed in year 2004: The time-depended, multimode, multiharmonic, fixed-field numerical code for the beam-field interaction in a coaxial gyrotron cavity, developed during previous periods, has been improved and has been used to study several coaxial-cavity configurations. In particular:

(i) The numerical instabilities of the code (triggered by the very dense mode spectrum of the ITER gyrotron) that were observed during the previous period have been studied exhaustively. Extensive tests and comparisons with the results obtained by the existing self-consistent code at the Helsinki University of Technology, Finland, showed that the two codes are generally in good agreement and that the unwanted numerical instabilities occur to both of them in more or less the same manner. (This work was done with the collaboration of Prof. Olgierd Dumbrajs from the Laboratory of Advanced Energy Systems, Helsinki University of Technology, Finland.) It was concluded that the instabilities can be eliminated, provided that the number of initial particle phases is chosen to be significantly larger compared to the usual numbers used up to now. Furthermore, for a given number of initial phases, one can obtain more accurate results by considering random initial phases instead of initial phases that differ by a fixed angle.

(ii) The code has been modified in order to incorporate the possibility of a magnetic field, which varies along the cavity axis. A Gaussian magnetic field profile has been adopted, to account for the fact that, in a realistic situation, the magnetic field diverges from homogeneity due to technological limitations. The background work to extend the code to be self-consistent rather than fixed-field has been continued. (This work will continue into next period.).

(iii) Additional simulations of the ITER gyrotron cavity have been performed, focusing on the effects of the kinetic energy spread on the start-up sequence of the modes (TE_35,19 – TE_34,19 – TE_33,19) and on the interaction efficiency. It was concluded that, for Gaussian spreads up to 4%rms., the mode sequence remains unchanged but, as expected, both the operating efficiency and the upper voltage limit of stability of the operating TE_34,19 mode decrease. Details of those simulations can be found in Annex VI.

(iv) Second-harmonic operation of coaxial cavities with corrugated insert has been further investigated in order to determine the parameters (cavity geometry, operating mode etc.) for a proof-of-principle experiment based on the facilities related to the ITER gyrotron project at FZK. For this task, a code for the calculation of the starting current of a mode using a realistic field profile has been developed. Pertinent results can be found in Annex VII and further calculations are expected to take place during next period.

 

2a5) Chaotic electron dynamics in gyrotron resonators

Background and Objectives: This is a new activity, formulated during the mobility mission of Prof. O. Dumbrajs from Helsinki University of Technology to NTUA. The main objective of the work is to analyse complex electron dynamics in gyrotron resonators in order to provide information about efficient operation of gyrotron devices.

Work performed in year 2004 (in co–operation with Helsinki University of Technology, Association Euratom-TEKES): Phase space analysis combined with a general Hamiltonian perturbation method was applied to the non-integrable dynamical system governing electron motion under the presence of an RF field in a gyrotron resonator. The following cases were considered:

(i) Electron interaction with one RF mode at the fundamental cyclotron frequency. The Canonical Perturbation Method has been applied in order to calculate approximate invariants of the motion, containing rich information for the phase space structure of the electron motion. Conditions for complex dynamical behaviour were obtained. The latter are connected with the drastic change of phase space topology and the deformation of certain KAM surfaces, which were shown to be essential for the operation of gyrotrons in parameter ranges of high efficiency. Moreover, information for the efficient operation of a depressed collector was provided. This operation requires an accurate knowledge of the spectrum of the electron rest energies. In principle this spectrum can become chaotic making the operation of a depressed collector meaningless or even dangerous due to reflected electrons. Investigations were performed under what circumstances (unfavorable resonator geometry and/or gyrotron operating parameters) this can happen. [See Annex VIII for more details or Y. Kominis et al., "Chaotic electron dynamics in gyrotron resonators", Phys. Plasmas 12, 043104 (2005).]

ii) Electron dynamics under the presence of one RF mode interacting at a higher harmonic of the cyclotron frequency has been studied in order to provide parameter ranges of high efficiency operation under such configurations. The application of the Canonical Perturbation Method has been extended and generalized to the case of resonant interaction at any harmonic of the electron cyclotron frequency. Moreover, the case of magnetic field tapering resulting in a varying frequency mismatch has also been included, and its effects on the resonant phase space areas have been shown. Also, the approximate invariants of the motion were used for calculating approximate electron distribution functions as solutions of the Vlasov (or Liouville) equation. Efficiency calculations have been given in terms of the approximate distribution function, resulting in expressions for the case of Dirac or Gaussian initial momentum distributions. It has been shown that the approximate distribution functions can also be used for the calculation of any other collective quantity of the electron beam. The full formulation and details on the results are presented in Annex IX. This work, combined with the study of electron interaction with multiple modes (work performed early in 2005), has been submitted for publication in Physics of Plasmas.

2b) Diagnostics and modelling of boundary layer plasmas and wall effects

Background and Objectives: Divertors remain the main option for handling plasma-wall interaction problems in operating and future fusion machines like ITER. The rather conflicting requirements to be fulfilled for efficient divertor operation ask for a detailed understanding of the involved physical and chemical processes. Important issues such as transient target plate power loads, pumping efficiency, target plate erosion, impurity migration and re-deposition are still very much under investigation. Langmuir Probes have proved to be a very useful diagnostic technique for divertor plasma studies and are now used extensively in most tokamak experiments. Despite the still existing interpretation problems and the unavoidable interaction with the plasma under investigation, Langmuir Probes can provide information with a spatial and temporal resolution very difficult to be obtained by other methods. The objective is to use and further enhance the fast scanning Langmuir probe system designed, constructed and operated by the Plasma Physics Laboratory at N.C.S.R. “Demokritos” for the ASDEX Upgrade divertor, to describe and eventually predict the detailed evolution of the SOL and divertor plasma, in conjunction with other diagnostic techniques and numerical simulations. (This is a multi-annual activity, performed in cooperation with IPP.)

2b1) Diagnostics and modelling of ASDEX-Upgrade SOL and divertor plasma

Work performed in year 2004: The divertor reciprocating Langmuir probe was used in conjunction with other diagnostics to study edge-divertor properties at ASDEX Upgrade. In total, the probe was used in 70 successful discharges. For various discharge configurations, probe measurements across both legs of the Div IIb divertor were made. More specifically:

(i) Electron temperature, density, ion saturation current and probe floating potential were measured for ohmic, L-mode, H-mode (up to 5 MW of additional heating power), lower single null and upper single null discharges, in both normal and reversed toroidal field. Comparisons with similar measurements made in the Div I and IIa configurations showed many similarities but also some differences. (For details see Annex X). In addition, discrepancies were observed between the separatrix position as determined from equilibrium reconstruction of magnetic measurements and the position suggested by the probe data, implying a possible (perhaps systematic) error of ±2 cm in the vertical positioning of the plasma. The reciprocating probe can therefore provide a useful tool for determining precisely the separatrix position in the lower divertor.

(ii) Several parallel measurements involving the divertor manipulator, the mid-plane manipulator and the fixed Langmuir probes were made. In the most successful cases the probe tip sampled flux tubes connected directly to the fixed divertor probes, permitting the observation of the reciprocating probe shadow on the fixed probe ion saturation signal (see Annex XI). A malfunction of the mid-plane manipulator during the last stages of the campaign did not permit this activity to be completed and some experiments will be repeated during the next campaign. Comparisons of probe experimental results with published SOL simulations were performed, showing several differences, especially in the density profiles of the inner SOL. Simulations using the exact discharge parameters are also now planned for next year (in collaboration with IPP).

(iii) Fast measurements of ion saturation current and floating potential fluctuations during ELMs were made. The ELM mitigation experiments were also successfully performed, results of which were submitted for peer reviewed publication. Oscillating ELM precursors (similar to what was observed in Div IIa measurements) were in fact found to be rotating main chamber MHD modes, uncorrelated with the ELM activity. Similar modes were observed in ohmic discharges, but the mechanism by which they interact with the divertor probe is unclear.

2b2) Modification of Langmuir probe hardware and software

Work performed in year 2004: In parallel to the use of the probe, several upgrades were made in the probe software and hardware. The driving software was updated to increase its usability during routine operation, making it now possible to finely adjust the motion parameters prior to each discharge (depending on discharge conditions). Concerning hardware upgrades, a new rail system has been designed and manufactured, and was installed on the manipulator during the summer vent (August 2004). This new system permits the easy removal of the manipulator from the port connection and greatly facilitates routine maintenance of the probe head during the campaign. A new vacuum control system was also prepared and installed during the vent.

2c) Equilibrium, stability and transport of fusion plasmas

2c1) Transport and chaos in fusion plasmas

Background and Objectives: One of the most complicated behaviours in dynamical systems is the motion of the turbulent fluid. In general, for a non-turbulent state, there exists some simple, usually linear, relation between the fluxes and the thermodynamic forces. The linearity leads, at least at macroscopic level, to predictability. On the contrary, the evolution of turbulent systems is unstable and two almost identical systems are very likely to evolve rapidly towards very different states. Turbulent systems are unpredictable. The sensitive dependence on initial conditions is a mark of «chaos». Our aim is to study chaotic dynamics and transport during critical situations in toroidal plasmas, such as sawteeth crashes, by utilizing the Lagrangian formulation of the guiding centre motion, which incorporates finite and neo-classical effects, as well as effects due to turbulence and fluctuating field components. The investigation is analytical as well as numerical. (This is a multi-annual activity, performed in cooperation with ULB.)

Work performed in year 2004: During this period, we considered the extension of the beam tracing code TORBEAM to include the propagation and absorption of more complicated beams. The main target for this period was to study non-Gaussian beams by superimposing a Gaussian mode and a higher-order Gaussian-Hermite term and follow the evolution of the beam in the plasma. The necessary modifications of the theoretical model to include these generalizations were completed. More precisely, the definition of the beam width was generalized in order to include the case of non-Gaussian beams, by using the notion of the rms width. Τhe new formulation, apart from the effect of the broadening of the profile due to the non-Gaussian character of the beam, includes also the effect of the different phase velocity between the modes. The strength of the effect of the phase difference depends on the amplitude of the electric field and on the order of the correction term. The formulation was tested for the case of an EC wave beam propagating without absorption in a tokamak plasma slab, and the results agreed with the theory. The new formulation was incorporated to the beam tracing code and test runs were made. The generalization of the method in order to be able to use more than one correction terms was completed. At the absorption layer the beam is expected to obtain very complicated shapes. The necessary steps were taken for the development of the theory for the description of the change in the beam shape due to the localized absorption in the plasma. Our main approach is the following: we continuously reconstruct the beam shape in terms of Gaussian-Hermite modes using a Fourier method as it propagates through the absorption layer. The propagation of the complicated beam will be followed using a sequence of shapes, and this sequence will be continued until the beam leaves the absorption region.

We also initiated the treatment of nonlinear absorption of wave beams in fusion magnetic configurations and tokamak plasma geometries, starting with the study of the plasma particles behaviour under the influence of an ECRH wave beam in a fusion plasma. We performed a self-consistent treatment of nonlinear wave-particle interaction (see Annex XII), for electromagnetic waves in the electron-cyclotron (EC) frequency regime and magnetized relativistic electrons. A closed set of nonlinear equations is used, which consists of the equations of motion under the wave field and the wave equation for the vector potential. The electron motions drive the evolution of the wave amplitude and frequency through the current density.

2c2) MHD turbulent transport in plasmas

Background and Objectives: Computational Fluid Dynamics (CFD) and turbulence modelling have been applied to various flow problems using codes, which have been developed and tested earlier. These codes are based on Navier-Stokes solvers in an Eulerian frame of reference, combined with Lagrangean particle dynamics. It is proposed to extend these CFD codes to solve problems of MHD turbulent transport, including eventually effects due to resistivity, for the purpose of studying numerically the turbulent diffusion of turbulent charged particles, using computational fluid dynamics techniques. In addition, as regards task (iv), earlier work to cast the MHD equations in Cellular Automata form, a formalism seen very efficient for 3D global simulations in confining magnetic topologies, is extended, in order for Cellular Automata to be used to study resistive instabilities on specific confinement devices. In addition, low-frequency electrostatic turbulence, driven by spatial gradients, is believed to be the dominant source of anomalous transport in magnetically confined fusion plasma. Special emphasis has been given lately on the properties of large-scale anisotropic flows generated by the drift-type turbulence, due to the critical role they play in the regulation of low-frequency drift instabilities and consequently of the levels of turbulent transport. The spontaneous generation of large-scale flows driven by electrostatic wave turbulence has been experimentally observed in various tokamaks and it is considered to be one of the hot subjects in the current fusion plasma research. (This is a multi-annual activity, performed in cooperation with ULB, IKET/FZK and U. of Uppsala.)

Work performed in year 2004: The following work has been done in the tasks of this activity:

(i) MHD natural convection of liquid metals in the presence of a magnetic field were studied as follows:

• Transient MHD natural convection: A numerical study was carried out of unsteady 2-D natural convection of an electrically conducting liquid in a laterally and volumetrically heated square cavity under the influence of a magnetic field. This flow is characterized by the external Rayleigh number, Ra_E, determined from the temperature difference of the side walls, the internal Rayleigh number, Ra_I, determined from the volumetric heat rate, and the Hartmann number, Ha, determined from the strength of the imposed magnetic field. Starting from given values of Ra_E and Ha, for which the flow has a steady unicellular pattern, and gradually increasing the ratio S = Ra_I/Ra_E, oscillatory convective flow may occur. The initial steady unicellular flow for S = 0 may undergo transition to steady or unsteady multi-cellular flow up to a threshold value of Ra_I depending on Ha. Oscillatory multi-cellular flow fields were observed for S values up to 100 for the range 10⁵ to 10⁶ of Ra_Estudied. The increase of the ratio S results usually in a transition from steady to unsteady flow but there have also been cases where the increase of S results in an inverse transition from unsteady to steady flow. Moreover, the usual damping effect of increasing Ha is not found to be straightforward connected with the resulting flow patterns in the present flow configuration. [See also Annex XIII and "MHD natural convection in a laterally and volumetrically heated square cavity", I. Sarris, S. Kakarantzas, A. Grecos and N. Vlachos, U. of Thessaly, accepted for publication in Int. J. Heat & Mass Transfer.]
• MHD Turbulent Transport in Fusion Plasmas (in collaboration with ULB): Study of MHD turbulence, using direct numerical simulations, in flows subjected to spatially periodic external magnetic fields. The influence of inhomogeneities in the kinetic or magnetic energies was investigated. The low magnetic Reynolds number approximation that has been successfully used in homogeneous turbulence was tested for these inhomogeneous flows. Also a study of turbulent dispersion in particle-laden turbulent channel flows was carried out. (See also Annex XIV.)

(ii) The static equilibrium and stability of 2D free convection flow in a square cavity was examined, in the presence of a uniform internal heat source and a uniform magnetic field that is perpendicular to gravity and parallel to an imposed temperature gradient. The finite element method is used for calculating the equilibrium and dynamic state of the system in the parameter space defined by the dimensionless numbers, Gr, Ha, Pr and S. The trapezoidal rule is used for time integration. Linear stability analysis is performed, by solving a generalized eigenvalue problem. The Arnoldi method is used for the calculation of eigenvalues with significant savings in storage and CPU time requirements. The base solution normally exhibits two recirculation regions. Stability analysis predicts a Hopf bifurcation to a new branch that is characterized by periodic alternation between an asymmetric and an almost symmetric configuration of the two rolls, with respect to the transverse centerline of the cavity. A neutral stability diagram is constructed for a range of values of Ha, Gr and Gr_i for liquid lithium, Pr = 0.0321. Internal heat generation, i.e. increasing S, enhances instability by decreasing the critical value of Grashof number, Gr_cr, beyond which the periodic solution prevails, whereas intensifying the magnetic field, i.e. increasing Ha, stabilizes the flow by increasing Gr_cr. Dynamic simulations confirm the above structure and recover the time constants predicted by stability analysis. As Gr increases or as Ha decreases symmetric arrangement of the two rolls is eliminated, the average temperature of the cavity is lowered and the steady flow configuration eventually loses stability. Subsequently, time periodicity sets in leading to more efficient heat removal, in comparison with the steady flow configuration (see Annex XV).

(iii) Modelling MHD flows in rotating cylindrical and spherical shells (in collaboration with FZK): The study of MHD flows inside a cylindrical shell with a rotating disk was continued for a wider range of Ha and Re values using the code CAFFA which was appropriately adapted and the electric potential equation implemented. A magnetic field was imposed parallel to the axis of the rotation. Linear and angular velocity fields, and distributions of the electric potential were obtained. The code was tested and corrected with valid solutions or experimental data and its capabilities were extended to MHD flows in rotating spherical shells. Also the MHD flow in spherical shells with rotating bodies (disks or spheres) was investigated. (See Annex XVI and "CFD simulation of rotating tangential layers (jets) in strong uniform magnetic field", D. Fidaros and N. Vlachos , U. of Thessaly, and L. Bühler, IKET/FZK, Collaboration Report 2004.)

(iv) A spectral numerical code was constructed which implements models of two-fluid plasma equations. The main purpose of this code is the numerical description of the temporal evolution of electrostatic plasma instabilities and the investigation of the turbulent generation of large scale sheared plasma flows. During year 2004, we focused on the investigation of the excitation and saturation of large-scale anisotropic flows as a result of the temporal evolution of the magnetic-curvature-driven electrostatic flute instability, taking into account finite ion Larmor effects. The formation of the streamer structures is attributed to the linear development of the instability, while the subsequent excitation of the zonal flows is the result of the non-linear coupling between linearly grown flute modes and particularly on the inverse energy cascade of the flute turbulence. Among others, it was shown that in the saturated state of flute turbulence, the streamer and the zonal modes-which determine the structure of large-scale plasma flows-become coupled exchanging energy with each other. Furthermore, the role of the ion temperature on the evolution of the flute instability was investigated, and it was shown, among others, that the dominant large--scale modes of the potential and the density may self-organize in different ways, depending on the value of the ion temperature. The results of our numerical investigations were submitted during 2004 and published at Phys. Plasmas 12(3), 032503 (2005) (see Annex XVII). Furthermore, during 2004 we investigated analytically the properties of the large-scale flows, which develop in and interact with an electrostatic turbulent environment of the flute type. For our purposes we used and analyzed a kinetic wave equation which describes the turbulence, coupled with the averaged fluid equations which describe the large scale flows. We showed that the resonant interaction between the variations of the mean flow and the turbulent spectra may lead to the stabilization of the large-scale flows. The oscillating frequency and the stability criterion for the zonal flows were analytically determined in terms of the turbulent spectra. In addition, we showed that further nonlinear evolution of the large flows may lead to the formation of stationary coherent structures in the transition layer between surfaces of different flow velocities, modifying significantly the transport properties of the turbulent plasma. The results of the investigations were submitted during 2004 and published at Phys. Plasmas 12(4), 042311 (2005) (See Annex XVIII).

2c3) Stochastic modelling of transport phenomena

Background and Objectives: This work concerns analytical and numerical investigations of kinetic equations, mainly of Fokker-Planck type, arising from stochastic modelling of the motion of charged particles in magnetized plasma. In particular, it encompasses the questions:
(i) Consistent approximation schemes beyond the quasi-linear theory;
(ii) Lattice kinetic models in magnetohydrodynamics and plasma turbulence; and
(iii) Boundary value problems relevant to fusion devices.
The objectives of (i) and (iii), to examine further effects on transport properties due to random components of the magnetic and/or electric field, especially concerning problems of anomalous diffusion. The objectives of (ii) are to develop MHD - Lattice Boltzmann algorithms to validate their ability to simulate MHD flows and plasma turbulence and finally, to solve in a hybrid manner, three dimensional dissipative MHD flows with particles in toroidal plasma. (This is a multi-annual activity, performed in cooperation with ULB and IPP-CR.) Furthermore, the Continuous Time Random Walk (CTRW) framework for particle transport is developed and applied to confined, turbulent plasmas. The aim in this approach is to model anomalous transport, to be able to understand the mechanism(s) that lead to anomalous transport, and to identify processes and configurations that suppress or increase transport.

Work performed in year 2004: The following work has been done in the tasks of this activity:

(i) Lattice kinetic models in MHD and plasma turbulence: The three dimensional lattice kinetic code (LMHD-3D) has been further validated by solving the 3-D Taylor-Green vortex problem. The computed temporal evolution of kinetic and magnetic energies and enstrophies show excellent quantitative agreement with corresponding results obtained by a pseudo-spectral code kindly provided by ULB. Also the LMHD-3D code has been upgraded by incorporating mpi subroutines to achieve high efficiency and speed. Even more, a thermal lattice kinetic module has been added in the code and some benchmarking has been performed. This is an important addition drastically enhancing the realm of applications, since now thermal MHD problems can be simulated. Finally, a speed up version of the classical LBM has been proposed on a theoretical basis. Experimental evidence is still needed. A more detailed description on all these issues are given in Annex XIX, where all related publications during 2004 are included. We have continued our collaboration with IPP-CR on basic issues related to lattice kinetic schemes. In addition, we have initiated a new collaboration with ENEA sulla Fussione, C. R. Frascati, in an effort to investigate the advantages (and disadvantages) of lattice kinetic schemes compared to typical computational approaches in fusion. Regarding educational activities it is important to note that we have organized and scheduled for the spring semester of 2005, a new graduate course (15 weeks) with title "Kinetic Theory and Microfluidics". The first half of the course is dedicated on issues related to kinetic equations (Boltzmann, BGK, Vlasov, Fokker-Planck), applied to several fields including plasma physics.

(ii) Turbulent transport in Tokamak plasmas: In the period in subject, the previously developed equations for the combined Continuous Time Random Walk (CTRW) in position and velocity space were brought to their final form, and ways to solve the equations were studied. Analytical solutions can only be derived for the asymptotic regime (large times, large distances, large velocities), and since Monte Carlo simulations clearly indicate that the transient phase is important and different from the asymptotic solution, the need for a numerical solution of the CTRW equations is given (heating, for instance, is a side-effect of CTRW and cannot be captured by asymptotic solutions). The CTRW equations are integral equations of the Volterra type, and we are under the way to solve them with the Nystrom method, which reduces the integral equations to algebraic equations. In a first approach, two separate integral equations for position and momentum space were formulated, and the equations were brought to an isotropic form, which reduces the dimensions of the problem. In the Nystrom method, the probability density functions for the position and momentum are followed in time on a set of prefixed position and momentum values along the radial direction. To achieve numerical stability, a number of integral transformations had to be applied. So far, we were successful to solve for the probability density function in momentum space, for the case where particles interact with localized electric fields, undergo trapping, and perform free flights (the free flight time is neglected). The probability density functions in momentum space basically coincided with the ones obtained from Monte Carlo simulations. A disadvantage of this two-equation-approach is that the velocity, when appearing in the position-space equation at a given time, must be replaced by the mean velocity at that time, which implies a loss of information on the random walk. We proceeded thus to the formulation of a self-consistent random walk equation, which is a single, combined integral equation for both the position and momentum variables, without the need of replacing some quantities by their mean in order to close the system. The problem of solving the equation numerically will be dealt with in the next year. The CTRW equation has now three independent variables (time., position, and momentum), its formal structure though makes it likely that it can be solved by a method similar to the one mentioned above. In parallel to the theoretical work, Monte-Carlo simulations of the CTRW were performed, continuing the work from last year, and analyzing more properties of the random walk, for the case of localized interactions with electric fields and trapping. An interesting result is that the particles may be sub-diffusive in position space, and in parallel they may be heated and accelerated in velocity space (power-law tails in the energy distributions are developed), as reported in a last years' annex with first results. The Monte-Carlo simulations also serve as a test for the numerical solution of the CTRW equations. In these simulations, the parameters, as far as they can be known at the time, are roughly adjusted to the situation of a typical Tokamak, A further adjustment of the parameters was not yet undertaken, as well as a comparison of different particle species and parametric studies, as announced in the tasks for the period in subject, since we put the stress on the development of the theoretical frame, which actually is a new approach, and which will allow to perform a much more in-depth analysis (parametric study, conditions for suppression-enhancement of transport) of the particle transport in confined plasmas. (See Annex XX for more details.)

2c4) Effects of superthermal particles during ICRH and ECRH

Work performed in year 2004: (This activity has been suspended from the work programme of years 2004-05: The principal investigator, Dr. A. Lazaros, is continuing work on this activity at FOM/FZJ, on a Euratom fellowship.)

2c5) Stationary MHD modes in magnetically confined plasmas

Background and Objectives: This is a long-term project (performed in cooperation with IPP) aiming at constructing equilibria and relaxed states of laboratory (and astrophysical) plasmas of fusion concern (e.g., plasmas of tokamaks and reversed-field-pinches) with flow or/and finite conductivity and investigating their linear and nonlinear stability. The understanding thus obtained can contribute to improving the current magnetic confinement systems and possibly developing new ones.

Work performed in year 2004:

(i) Construction of axisymmetric equilibria with toroidal flow and anisotropic resistivity: The MHD equilibrium states of an axisymmetric plasma with toroidal flow and anisotropic resistivity are governed self consistently by an elliptic differential equation for the poloidal magnetic flux function, a Bernoulli relation for the pressure and two relations involving the parallel and perpendicular, to the magnetic field, resistivity components. Unlike the case of equilibria with parallel flows [G. N. Throumoulopoulos and H. Tasso, Phys. Plasmas 10, 2382 (2003)], the current density surfaces coincide with the magnetic surfaces, and electric fields E perpendicular to the magnetic surfaces which are of relevance to the improved confinement regimes in tokamaks, are possible. Exact solutions with roughly collisional parallel and perpendicular, to the magnetic field, resistivity profiles, i.e. profiles having a minimum close to the magnetic axis, taking very large values on the boundary and such that the perpendicular being larger than the parallel have been constructed. Also, the flow impact on the profiles of the parallel and perpendicular, to the magnetic field, resistivity components, E, and the toroidal current density has been evaluated. Details on this study are presented in Annex XXI.

(ii) Impact of flow on the existence of tokamak equilibria with reversed central current density and construction of toroidal axisymmetric equilibria with reversed magnetic shear with particular attention to current-hole configurations: Multitoroidal configurations of an axisymmetric toroidal plasma have been studied as equilibrium toroidal-flow eigenstates within the framework of ideal MHD theory. The study includes incompressible plasmas and "compressible" ones with constant temperature but varying density on the magnetic surfaces. The flow in conjunction with the toroidicity can result in configurations with non-nested magnetic surfaces similar in magnetic topology with the static (no flow) ones proposed in connection with the "current holes" in tokamaks [A. A. Martynov, S. Yu. Medvedev, L. Villard, Phys. Rev. Lett. 91, 085004 (2003)]. In addition, for magnetic topology of non nested magnetic surfaces the current density can reverse in the core region. On the basis of exact equilibrium solutions constructed it turns out that for "compressible" plasmas the maximum variation of the density on a magnetic surface is as large as 5% and it is equal to the temperature variation on a magnetic surface for incompressible plasmas. Also, the impact of the flow on the Shafranov shift has been evaluated. An extensive summary on this work is presented inAnnex XXII.

2c6) 3-D pellet modelling

Background and Objectives: The 3-D pellet modelling activity is performed in collaboration with IPP. The objectives of this activity are to develop multi-dimensional pellet codes for pellet-plasma interaction studies. The multi-dimensional codes are: the development of a "2D+1" pellet-plasma code and the implementation of pellet module in a 3-D MHD code, the M3D code developed by Princeton . The "2D+1" code is 2-D resistive MHD in the poloidal plane plus 1-D Lagrangian in the toroidal direction. The current version of the "2D+1" has only one axial magnetic field component B_z.

Work performed in 2004: Work continued on the implementation of multi-cell in the third direction of the "2D+1" code. The thermal diffusion equation, parallel to the magnetic field, is currently being implemented and tested on a Eulerian grid. A large number of scenario runs were performed with the "2D+1" code, with and without magnetic field gradients. With magnetic field gradients the magnetic field applied (B_z) is linear and varying only in the x-direction and constant in time. The scenario runs, which were performed with magnetic field gradient, had no moving neutral particle source, but instead a circular high density low temperature cloud (T_c = 1.5 eV, N_c = 5×10²³ m^–3), initially stationary, is placed in the centre of the numerical grid, the rest of the numerical grid is filled with an initial hot (T_e = 1 keV, N_e = 5×10¹⁹ m^–3) plasma, and the system was allowed to evolve for some hundreds of microseconds. Drift velocities of more than 1000 m/s are obtained in these scenario calculations.

Scenario runs with moving neutral particle source were also performed. One interesting feature in the computations with moving neutral-particle sources, is the formation of a thermal shock ahead of the neutral particle source. In order to understand these thermal shocks, a number of shock tube tests were performed (see Annex XXIII).

The shock tube tests were performed by modifying: the initial conditions, the boundary conditions, and the expansion in the z direction of the "2D+1" code, such as to have only 1-D flow. Shock tube tests with magnetic field were also performed, and found to be in good agreement with the Rankine-Hugoniot conditions. Using this modified 2-D code which has, 1-D flow only, the expansion and drift of a high density (low temperature) plasmoid in a low density high temperature plasma with a magnetic field, which has an initial field gradient, was also studied.

The collaboration with HAS was terminated because of lack of manpower; from the ADAS database interpolating polynomials have been obtained for the radiation of hydrogen, but a radiation module has not been implemented in the "2D+1" code yet.

2c7) Energy transfer due to wave - particle interaction

Background and Objectives: In recent theoretical developments (E. A. Evangelidis & G. J. J. Botha, in press) for the transfer of energy from waves to particles (and vice versa), analytical expressions have been obtained for the interaction of elliptically polarized waves with hot particles. It has been shown that the currents supported by a magnetized plasma can be classified as 0, ±1, ±2 modes and further that only the 0 and ±2 modes are available for transfer of energy. Thus, the transfer rates have been calculated using certain identities on summation of Bessel function products (derived earlier by EAE) and some integral expressions (appearing in the open literature for the first time) suitable for thermalized plasmas, which, simplify the algebra by a considerable amount. Furthermore, transfer rates have been calculated for the interaction of waves with non thermal plasmas, a common occurrence in solar plasma and fast particles in tokamak plasmas, upon using kappa – distribution functions. (Work performed in collaboration with JET.)

Work performed in 2004: Preliminary analysis was performed on existing experimental data on fast particles due to their interaction with Ion Cyclotron waves. (This activity started in mid-November 2004 and is to be continued into next period.)