## 2.PHYSICS PROGRESS 1999-2000

## 2a) Beam-wave interactions and high-power rf generation

### 2a1) Non-cylindrical *e*-beam for quasi-optical gyrotron

The quasi-optical gyrotron was initially [Sprangle, Vomvoridis and Manheimer, Phys. Rev. A 23, 3127 (1981)] visualised to employ a sheet beam, in order to take full advantage of its potential as a high-power microwave source for fusion plasma heating. Relevant experiments at CRPP [Tran, oral communication] on sheet beam propagation in a cylindrical duct have indicated a deformation of the beam cross-section to the shape of a spiral. This deformation is attributed to the E×B drift in the electric field produced in the azimuthal asymmetry of the configuration. This activity has as objective to identify the appropriate structure of the beam tunnel, so as to prevent the deformation of the sheet beam. (*This is a multi-annual activity, continued into the following periods and performed in cooperation with CRPP.*)

During the period in subject the problem has been formulated and the 2-D elliptical geometry has been investigated. More details are presented in Annex I. At first the problem, consisting of Poisson/Laplace equations and electron dynamics, with the additional condition that the beam boundary is an equi-potential surface (in order to eliminate deformations caused by the E×B drift), has been formulated in two dimensions (i.e., with ∂/∂z = 0). This is meant to be a first approach to describe a beam tunnel with very weak axial variation. The formulation refers to arbitrary rectilinear co-ordinates. Subsequently, the 2-D formalism has been applied to elliptical co-ordinates, to describe an electron beam with elliptical cross section (which, in the limit of large eccentricity reproduces a sheet beam). The dimensionless Poisson/Laplace equations have been solved analytically and the equi-potential lines (possible shape of conducting boundaries) and the field lines (possible shape of open boundaries) have been obtained to specify appropriate shape of tunnel boundary. Finally, the limit of a sheet beam is generated from the elliptical cross section and the results have been discussed in relation to their applicability to the design of a beam tunnel suitable for a quasi-optical gyrotron. This discussion (limited by the assumption of so weak an axial variation, so that the approximation ∂/∂z = 0 be adequate) indicates that indeed such designs have the potential to produce gyrotrons well in excess of 1 MW.

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

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). The objective of this activity is to prepare a self-consistent electrostatic code in three dimensions, for use in gyrotron beam tunnel studies. (*This is a multi-annual activity, continued into the following periods and performed in cooperation with CRPP.*)

During the period in subject the logic for the structure of the code has been addressed and certain background codes have been developed and tested. In particular, at first the code structure has been defined, including the procedure for the user accessing the database. Subsequently, the most common topologies for the geometry of a beam tunnel have been identified and the software has been developed and tested for definition of the boundaries, along with the appropriate conditions (of Dirichlet or Neumann type). This is accomplished by joining contours to form segments of the tunnel and subsequently joining adjacent contours with the help of suitable interfaces. Finally, a 3-D mesh generator has been developed and applied to typical geometries for the beam tunnel. In the generator the flexibility has been provided for the mesh size to be adjusted, so that continuity is present across the interfaces between adjacent segments, as well as to satisfy any particularities of any part of the beam tunnel (e.g., the necessity for a denser mesh in the region occupied by the beam). (More details are presented in Annex II.) In addition, the variational principle has been applied to construct a 3-D Poisson solver suitable for application of the finite element method. The algorithm has been tested in a number of vaccuum geometries, for which analytical solutions are available. Finally, a particle dynamics solver (employing the Runge-Kutta algorithm) has been initiated to advance the electrons and update the beam density.

### 2a3) Electromagnetic code for beam-tunnel spectrum

This task constitutes an extension of that performed under cost-sharing actions (prior to the conclusion of a Contract of Association between Euratom and the Hellenic Republic) and has as its objective to study the TE and the hybrid eigenmodes in a non–periodic corrugated waveguide and to translate existing numerical codes (developed in FORTRAN) to a more user-friendly language with graphics environment (in C) for the TE and TM modes in a periodic corrugated waveguide. (*This is a multi-annual activity, continued into the following periods and performed in cooperation with CRPP.*)

During the period in subject, a theoretical technique for determining the dispersion relation, the electromagnetic field components and the quality factor of a dielectric-loaded non-periodic corrugated waveguide has been developed (see Annex III for more details), for the case of the hybrid waves. The Floquet theorem is used to express the field distribution in the vacuum region, while an eigenfunction expansion is employed in each dielectric region. Applying the appropriate boundary conditions at the interfaces leads to a system of two infinite sets of equations, which can be solved numerically by truncation. This represents a fully general analysis, which includes the special cases (a) of a periodic corrugated waveguide (by selecting just one dielectric ring) and (b) of azimuthal independence (i.e., pure TE or TM waves). The corresponding relations for the last two kind of modes are obtained by choosing no angular dependence (m = 0) and setting the corresponding expansion coefficient of the axial electric or magnetic field equal to zero.

In addition (Annex IV), a numerical code has been developed in Microsoft Visual C++ 6.0 by use of the Microsoft Foundation Class Library. Its purpose is to supply a powerful tool for the calculation of the characteristics of a gyrotron beam channel. It offers the capability to calculate the dispersion relation, the quality factor, the electromagnetic field distributions and the energy for all kind of modes that in principle can propagate in a gyrotron beam tunnel (i.e., TM, TE and hybrid). It covers both periodic and non-periodic geometries with as many as 32768 dielectric rings. The code has been developed in such a way that it offers fast calculations with accuracy, which can be chosen by the user. Special attention has been made in making the program as user friendly as possible. All parameters are organized in a comprehensive way into dialog boxes, whereas the results can be displayed as text or as a graph. Using a multi-document/multi-view architecture, the code allows multiple documents to be opened, but most of all it permits different views of the results. This gives the possibility to present the results as text and as graph at the same time. The friendliness of the program is completed by allowing the storage of all the parameter values to a file, their retrieval from a previously saved file, and by explaining the use of the program with the aid of a help file.

### 2a4) Coaxial and harmonic gyrotrons

The operating mode has been selected using the method of normalised variables and imposing design limitations relevant to CW operation, such as: cavity outer-wall loading <1 kW/cm^{2}, coaxial insert wall loading <0.15 kW/cm^{2}, emitter current density <4 A/cm^{2} and operating beam current less than half of the limiting beam current. The TE_{-35,17} mode has been selected for the operation of the gyrotron under design. According to the approach of normalized variables, the mode is able to deliver the requested power of 2 MW at beam voltage 86 kV satisfying all the CW-relevant design criteria. A coaxial resonator has been designed to oscillate in the design operating mode at 170 GHz. The cold-cavity field profile of the TE_{35,17} mode has been obtained and the diffractive quality factor has been calculated at 1053. Gyrotron interaction calculations are obtained in the cold-cavity fixed-field approximation. The efficiency was limited to 34% maximum, but this was due to the low quality factor of the cold-cavity field. Starting current curves has been obtained in terms of the magnetic field and the beam voltage. In addition, results have been obtained in the stationary self-consistent field approximation. A beam of 10.3 mm radius and 71 A with a velocity ratio of 1.3 is able to produce in the designed resonator at 86 kV oscillations at 170.01 GHz with a RF power of 2358 kW and electronic efficiency 38.6%. Assuming total losses of 15%, the output power is expected at 2000 kW with efficiency 32%. The dependence of the output power on various parameters as the velocity ratio, the beam current, the magnetic field and the beam voltage has been studied. Finally, the time-dependent, self-consistent calculations have been performed. The operating mode has been found stable at the design point in single-mode approximation. All the modes able to be excited by the beam have been studied in single-mode approximation. The results have been used to investigate the stability of the operating mode and the behaviour of the tube during the start-up in a multi-mode environment. Indeed, the TE_{-35,17} mode has been found stable at the operating voltage of 86 kV in the presence of all competitors. But the study of the behaviour of the tube during the start-up procedure (rise of the beam voltage to the nominal value) has shown that the designed operating mode is not able to be excited and reach the high-power region. Instead, it is the satellite TE_{-34,17}, which is excited after the TE_{-36,17} losses oscillations at about 75 kV. In addition, the TE_{-34,17} keeps stable oscillations at 86 kV at a power level of 1200 kW with electronic efficiency 20%. The reasons for this jumping from the TE_{-36,17} to the TE_{-34,17} without passing through the TE_{-35,17} as it was designed are not yet known. Comparison with previous designs indicates only the relatively low quality factor of the operating mode as a possible reason. This is currently under investigation and if it will be proven correct, a new design must be performed with a new operating mode. (*Work performed, in parallel to the work of the Association, under contract ERB 5004 CT98 0019 and in co-operation with FZK.*)

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

Divertors remain the main option for handling the plasma-wall interaction problems in the operating and future fusion machines like ITER. The rather conflicting requirements to be fulfilled by a successful divertor operation ask for a detailed understanding of the involved physical and chemical processes. The use of Langmuir Probes proves to be a very useful diagnostic technique for divertor plasma investigations and are now used in all 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 fast scanning Langmuir Probe System (LPS) for investigations of ASDEX-Upgrade divertor area plasma has been designed, manufactured and set in operation under cost-sharing contracts (prior to the conclusion of the Contract of Association). [*It has been a cooperative project between NCSR "DEMOKRITOS", Institute for Nuclear Technology–Radiation Protection and Max-Planck-Institut fur Plasmaphysik (IPP) EURATOM Association.*] The LPS has been designed for the open DIV-I configuration of ASDEX-Upgrade and has had to be modified for the DIV-II, LYRA configuration, loosing its 2D scanning capabilities. Full stroke radial profiles of the electron temperature, Te, plasma density, Ne, floating potential, Vfloat and plasma flow (Mach number) have been recorded for ohmic (OH) and low power (2.5 MW) neutral beam injection (NI) heated discharges. In these cases a probe stroke across the entire divertor was possible, allowing direct comparison of plasma parameters in the two divertor legs and investigations of the private flux region. In addition, strong divertor asymmetries, complex plasma flow patterns (including the private flux region below the X point) and changes in the power flow to the divertor targets, depending on ion gradB drift direction, density and divertor geometry, have been observed and reported. First comparisons with the predictions of B2, using a reduced grid, for the OH shots have been successfully made and a significant effort has been made to include the drift and current terms in the multifluid version of the B2 SOL simulation code.

In view of the significance to the European Fusion Program of activities like the extension of the power withstanding capabilities of the LPS, the recovery of its 2D scanning capabilities, the provision of supplementary channels for fluctuations measurements, as well as the continuation of the modelling effort in understanding the SOL Physics and Langmuir Probe operation, the objectives of the continuation of this activity are (a) to describe and, eventually, predict the detailed evolution of SOL and divertor plasma, using the capabilities of the fast scanning Langmuir Probe System operating in ASDEX-Upgrade's divertor area, in conjunction with other diagnostic techniques and the potential of the SOL simulation code B2 (in its latest form including drifts and currents), and (b) to investigate, using the high temporal resolution of the system, some fast transient phenomena like ELMs, VDEs, disruptions and unipolar arcs, as they are seen in the divertor area. (*This is a multi-annual activity, continued into the following periods and performed in cooperation with IPP.*)

During the period in subject and in close collaboration with IPP, Garching, both the experimental and theoretical aspects of the ASDEX-Upgrade SOL and divertor plasma behaviour have been addressed. In particular, as regards the experimental part of the activities, the design and construction of the fast scanning LPS reached its final stage. This system, being capable of accessing both divertor legs, represents a valuable diagnostic tool for the investigation of the AUG divertor plasma. Profiles of ion saturation current density, electron temperature, electron density, floating potential and Mach number have been recorded, during ohmic and medium power neutral beam heated discharges. Emphasis was given on the influence of the latest DIV-II divertor geometry, as well as, the ion grad B drift direction on the divertor parameters and the power flux to the targets. These observations furnished the impetus for further numerical investigations, as described in the following. The collected results appear in N. Tsois, C. Dorn, G. Kyriakakis, M. Markoulaki, M. Pflug, G. Schramm, P. Theodoropoulos, P. Xanthopoulos, M. Weinlich, the AUG Team, "A fast scanning Langnuir Probe system for Asdex-Upgrade divertor", J. Nuclear Materials 266-269, 1230-1233 (1999). In addition and on the theoretical part, the multifluid code B2-solps5.0, for the numerical investigation of the AUG edge plasma, was substantially extended. Specifically, the equations describing the evolution of the plasma parameters were improved with the inclusion of diamagnetic flows caused by the presence of the electric field and the magnetic field inhomogeneity (drifts). An ad hoc model for the anomalous radial current, affecting the energy equations, was incorporated, as well. These new features were successfully benchmarked, in conjuction with an additional neutral hybrid model. Various scenarios, switching on/off drift effects and impurities were run, reproducing surprising well the experimental results, as obtained from the LPS measurements. The collected work appears in R. Schneider, D. Coster, B. Braams, P. Xanthopoulos, V. Roszhansky, S. Voskoboynikov, L. Kovaltsova, H. Buerbaumer, B2-solps5.0, "SOL transport code with drifts and currents", Contrib. Plasma Phys. 40, 328-333 (2000).

## 2c) Equilibrium stability and transport of fusion plasmas

### 2c1) Transport and chaos in fusion plasmas

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». Accordingly, as the objective for this activity has been set to study (analytically as well as numerically) 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. (*This is a multi-annual activity, continued into the following periods and performed in cooperation with ULB.*)

During the period in subject, as is elaborated in Annex V, the relativistic analysis has been performed for the physical system of the interaction of electrons with a single electromagnetic wave that propagates at an angle with respect to a uniform external magnetic field. In order to study the transition to stochasticity, the basic periodic orbits of the system have been found and their evolution followed, as the parameters of the system change. The regions of stochastic behaviour are visualised, using the well-known Poincare surface of section (PSS) method. From the numerical results it has been found that the energy band structure defines the regular motion. Indeed, when the motion is regular, the energies of the accelerated electrons stay within a narrow band, but when stochastic motion enters into play (i.e. when α , ε or the initial energy increase) this band broadens. This information gives a quantitative approximation of the threshold for stochastic motion, either for the normalised amplitude ε or the angle of propagation α . It can be inferred that, since the energy can be directly measured, the energy diagrams are a very elegant way of contrasting the accelerated electron dynamics to the experimental reality.

In addition, the stability and instabilities of periodic orbits of autonomous Hamiltonian systems have been studied. The different types of instabilities have been classified, using terminology suitable to many degrees of freedom. All possible transitions between different stability types are registered, as well as the equation for the transition hypersurface, using its dimension as an indicator of the probability for the transition to occur. Specific application has been made to well-known cases with two and three degrees of freedom. Finally, the different stability regions in three-dimensional parameter space are studied in detail, for a Hamiltonian system with four degrees of freedom.

### 2c2) MHD turbulent transport in plasmas

Computational Fluid Dymanics (CFD) and turbulence modelling have been applied (prior to the conclusion of the Contract of Association) to various flow problems using codes which have been developed and tested in the Fluid Mechanics Laboratory of the University of Thessaly. These codes are based on Navier-Stokes solvers in an Eulerian frame of reference, combined with Lagrangean particle dynamics. The general objective is to extend these CFD codes to solve fusion-relevant problems of MHD turbulent transport, including eventually effects due to resistivity, and to compare the results with those using LES (large eddy simulation) techniques, as well as by direct numerical simulation. Charged-particle turbulent diffusion is also to be considered, with various boundary conditions in simple and toroidal geometries. In addition, the use of particle image velocimetry (PIV-under development in the Laboratory) has been identified as a means to study simulated flows pertinent to specific aspects of turbulent diffusion in random fields. The specific objective is to study numerically the turbulent diffusion of turbulent charged particle, using computational fluid dynamics techniques, and to examine the use of laser flow diagnostics. *(This is a multi-annual activity, **continued into the following periods and performed in cooperation with ULB.)*

During the period in subject, the work in this field was mainly focused in adapting the CFD code DIAN3D to MHD flows. This source code had been developed in the Fluid Mechanics Laboratory of the University of Thessaly and it is basically a Navier-Stokes solver using the finite volume approach. The code can simulate steady or time-dependent laminar or turbulent flows in Cartesian or cylindrical coordinates. For turbulent flows it uses the classical two-equation *k-ε *turbulence model. It also can handle particle dynamics using the Lagrangean approach. In particular, for the case of an incompressible fluid, the code has been extended to MHD in three steps. First, assuming known the magnetic field, the effect of Lorentz force to the fluid flow has been considered. Next, assuming known the flow, an algorithm has been developed for the magnetic induction equation. Finally, the appropriate modifications were introduced to treat the coupled system of the Navier-Stokes and the induction equations. Moreover, using the so-called *k-ε *model, the effects of fluid turbulence are being studied. In addition, to test the modified DIAN3D code, computations have been performed for the two-dimensional flow between essentially semi-infinite parallel conducting plates. With the external magnetic field perpendicular to the plates, for the fully developed flow, the velocity profile and the magnetic field must coincide with the expressions obtained analytically for the Poiseuille-Hartmann flow. This is a means to check the various steps of the algorithm and the precision of the computation, which of course depends on the size of the grid. It should be noted that one of the aims of this research is to have a code, which generates reliable results in short (computation) time.

### 2c3) Stochastic modelling of transport phenomena

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) Relation of stochastic models to theories based on statistical mechanics; and (iii) Boundary value problems relevant to fusion devices. The particular objective is to examine further effects on transport properties due to random components of the magnetic and/or electric field, especially concerning problems of anomalous diffusion. (*This is a multi-annual activity, continued into the following periods and performed in cooperation with ULB.*)

During the period in subject, the Fokker-Planck equation in the weak coupling limit (which governs the probability distribution of a test particle in a plasma) was modified, by introducing in the collision term the effects of an external magnetic field. The results are well understood for motions perpendicular to the field lines. However, the statistical mechanical formalism leads to inconsistencies for the motion along the lines. This problem has been studied and clarified in the case, where the effect of the bulk plasma is modelled by a Gaussian ("coloured") noise.

### 2c4) Non-resonant stabilisation of sawteeth and tearing modes by superthermal particles (ions and electrons) during ICRH and ECRH

During the period in subject, the observed suppression/enhancement of tearing modes [Sing et al., Phys. Fluids B 5, 3239 (1993)], [Hoshino et al., Phys. Rev. Lett. 69, 2208 (1992)], during perpendicular ECRH was addressed. The characteristic feature of ECRH is the strongly localised power deposition at the resonant layer. In that respect ECRH essentially provides a localised source of superthermal electrons with a very peaked density profile at the resonant layer. The concept of the non-resonant power-transfer, from the superthermal electrons to the m = 2 mode, was applied (Annex VI) and the theoretical predictions (that the mode would be always suppressed if the resonant layer is outside the q = 2 surface while the mode would be either enhanced or suppressed when the resonant layer is inside the q = 2 surface) are qualitatively consistent with all of the available experimental results. It was also shown quantitatively that the required superthermal electron density (for the suppression of the tearing mode) is two orders of magnitude below the thermal electron density.

The proposed mechanism for superthermal electron generation at the resonant layer during ECRH is the well-known "avalanche" effect (Annex VII), in which the superthermal electron density grows exponentially. It was found that the superthermal density growth time depends on the fifth power of the thermal electron density, consequently the non-resonant interaction would be practically effective (for the suppression of tearing modes) in low-density plasmas.

In a second application of the theory the suppression of sawteeth oscillations, which was observed during on-axis ICRH in JET [Campbell et al., Phys. Rev. Lett. 60, 2148 (1988)] and TFTR [Phillips et al., Phys. Fluids B 4, 2155 (1992)], and also during tangential NBI in JT-60U [Kramer et al., Nucl. Fusion 40, 1383 (2000)], is attributed (Annex VIII) to the non-resonant power-transfer from the superthermal ions to the m=1 mode. In the present interpretation the sustainement of the "monster" sawtooth (through-out the ICRH pulse) is observed when the power-transfer to the m=1 mode remains negative/stabilising. On the contrary the sawtooth crashes (during the ICRH pulse) when the power-transfer eventually becomes positive/destabilising. For the needs of this study, theoretical models have been also presented for the development of a finite parallel electric field (in the presence of a magnetic mode) and the "effective diffusion" of passing ions across a magnetic island which is essentially a consequence of their collissional thermalisation.

### 2c5) Stationary MHD states in magnetically confined plasmas

This activity is a continuation to the work performed under earlier cost–sharing actions (prior to the conclusion of the Contract of Association). Its objective is to advance the status of understanding of the equilibrium, relaxation and stability properties of magnetically confined (i.e., fusion) plasmas. (*This is a multi-annual activity continued in successive periods and performed in cooperation with IPP.*)

During the period in subject, helically symmetric and generally symmetric (two-dimensional) ideal MHD equilibria with incompressible flows have been studied (Annex IX). In brief, the flux integrals have been identified and led to the reduction of the MHD equilibrium equations to a set of algebraic and differential equations. Then, analytic solutions for physically plausible cases have been constructed. In addition, the problems of existence of equilibria with purely helical flows, and isothermal magnetic surfaces have been examined.

Furthermore, the magnetohydrodynamic equilibrium states of an axisymmetric toroidal plasma with finite resistivity and flows parallel to the magnetic field lines are investigated (Annex X). The appropriate differential equation for the poloidal magnetic flux function is obtained and its solutions are investigated for a conductivity depending on the flux only or on the flux and the radial distance. The main conclusion has been that for the system under consideration, the existence of equilibrium depends critically on the spatial dependence of the conductivity. Analytic tokamak equilibria with sheared flows and radial electric fields similar to those observed in the L-H transition are constructed (Annex XI), while the extension of the work to include gravitational forces is also made (Annex XII). Finally, a sufficient condition for the linear stability of nonautonomous dissipative mechanical systems with circulatory forces, which is of importance for plasma equilibria with sheared flows, is derived (Annex XIII).

### 2c6) Vapour shield phenomena during hard disruptions in tokamak plasmas

A 2-D numerical code has been developed under a cost-sharing contract (prior to the conclusion of the Contract of Association), for calculating the time evolution of radiating vapour clouds over ablating solid surfaces (i.e., the graphite divertor plates), subjected to magnetically confined energetic plasma particles. The objective is to extend the code, inter alia in three dimensions, for application to studies of tokomak plasmas. (*This is a multi-annual activity, continued into the following periods and performed in cooperation with IPP.*)

During the period in subject, an initial evaluation has been launched, of the 3-D MHD code (M3D), developed by Princeton and Courant Institute. This evaluation aims at assessing, how the various pellet ablation modules, developed at European institutions (including FORTH) can be transferred to and used by the M3D code. Agreement has been reached on an interface module and the ablation physics modules. In addition, work has been done on the implementation of the 3rd magnetic field component in the 2-D surface erosion and vapour shield code.

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