## 2.PHYSICS PROGRESS 2001

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

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

During the period in subject, the work of the previous periods was continued, by taking into account the presence of an axial nonuniformity in the system. As is typical in gyrotron beam tunnels, the nonuniformity ∂/∂z in the magnetostatic field **B** was assumed to be weak, so that an adiabatic, WKB-type approximation is applicable. The solution for the electrostatic potential Φ_{0} when ∂/∂z = 0 was taken as a first estimate, and the problem was formulated for the case ∂Φ_{0}/∂z ≠ 0 [caused by the dependence of the beam density on **B**(z)]. It turns out that the first derivative ∂Φ_{0}/∂z does not contribute and the corresponding correction Φ_{1} vanishes identically. The leading contribution comes from the second derivative, ∂^{2}Φ_{0}/∂z^{2}. The equations for the corresponding correction Φ_{2} have been obtained and solved analytically. The solutions have in general the same structure, as in the case of axial uniformity (see report for year 2000) The most pronounced new feature in the results is that the electrostatic complex potential depends on the complex elliptical coordinate θ = ρ + jψ in terms of not only cosh(2θ), but also cosh(4θ). This has as a consequence that cross sections of the beam tunnel have a truly three-dimensional shape (with no similarity of the shape at different z positions, to assist in the machining of the beam tunnel).

In addition, some effort has been devoted in investigating, whether alternate geometries can be treated analytically. If one starts from a rectangular beam cross-section (to generate subsequently the limit of a sheet beam), the zero order solution for the electrostatic potential inside the beam can be represented by an infinite series. The series has very slow convergence (primarily due to the presence of the corners) to make it useful for a staring point for the analytic continuation beyond the domain of the beam. Even more complicated appears to be the situation with a fragmented beam, e.g., consisting of an array of ellipses. The difficulty here arises from the presence of the adjacent beamlets, which destroy the symmetry and do not permit an analytical treatment. It appears that such configurations can be treated only numerically (but even then, such problems of synthesis are not guaranteed to have a solution).

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

During the present period, the code ARIADNE (Annex I) has been brought to an operational (although not yet final) state. At first the particle pushing subroutine was created and implemented. A Runge-Kutta algorithm was employed, which follows the electron motion in the fast time scale (i.e., the time step is a fraction of the gyroperiod). [Although this choice is slower as far as electron dynamics is concerned, it has the advantage that the intermediate positions of the electrons are denser. Therefore, to represent the electron beam density and to calculate the electrostatic potential, a smaller number of simulation electrons is needed. Because of this trade-off, this choice of particle pusher does not affect the overall speed of the code.] The subroutine has been tested, by comparing the electron trajectories both with analytically available results and with the results from DAPHNE at CRPP.

The code has been supplied with a wide spectrum of choices for the introduction of the externally applied magnetostatic field. They include:

(i) Specification via table data in a suitable file.

(ii) Specification of the current carrying filaments and application of Biot-Savart law.

(iii) Specification of the properties (position, size, current) of cylindrical coils and calculation of the resulting field from superposition of the model field from a circular filament. [The latter choice is particular suitable to typical gyrotrons.] It is noted that the axis of the magnetic field does not need to coincide with that of the beam tunnel.

Improvements have been introduced in the structure of the code, such as, the management of the computer memory has been transformed from static to dynamic, many libraries have been incorporated for the formal manipulation of the mesh elements, the storage and the solver of the system etc.

In addition, detailed comparisons have been made between the results of the code and those from DAPHNE, in systems with azimuthal symmetry (to which the applicability of the latter code is limited). These comparisons (together with the tests performed earlier on each individual segment of the code) have established its reliability.

Finally, additional methods have been defined for the introduction and manipulation of the entities in the programme database. The earlier forms of graphical representation of the output have been improved and some new ones (e.g., representation of the electron beam evolution, in the co-ordinates of the user's choice, graphical presentation of the electric field direction, etc.) have been incorporated. Also, preliminary consideration has been given on the desired structure of the graphical user interface, primarily as regards data entry (in particular, the three-dimensional geometry) and a detailed analysis of the relevant steps has been prepared.

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

During the period in subject (see also Annex II and Annex III for more details), at first the numerical codes for the beam-tunnel spectrum (developed in the previous periods) have been updated, taking into account the suggestions of CRPP, after the mobility to CRPP in January 2001. In particular, we have finished the numerical codes under Windows environment for the case of periodic and non-periodic corrugated waveguides for all kind of modes. We have checked our codes with the results obtained by the well-known code CASCADE, as well as with results provided to us by Dr. S. Alberti of CRPP. In addition, we have compared the results obtained by our code with those obtained by an already established numerical code (Mafia). [Micha Dehler of Paul Scherrer Institute has kindly made such results available to us.] Finally, a poster has been presented in the 26th Int. Conf. Infrared and Millimeter Waves, Toulouse France, 10-14/9/2001.

In addition, continuing the work of the previous period, substantial effort has been invested in developing the numerical code in C, under Unix environment, for the non-periodic corrugated waveguide, which include all types of waves. At first, we have transformed the numerical codes from Windows to Linux and now we are making the necessary test runs. [The transformation of the codes in C under Unix environment will be completed by the first few months of 2002. The problem is that many systems with Unix environment do not support all the libraries included in the codes and therefore special attention must be given in the transformation and the compilation of these codes.] Up to now, we have written the appropriate graphic user interface for the TE, TM and Hybrid waves in a non-periodic corrugated waveguide with real dielectric permittivity in the corrugations. The selected language under Unix environment is Tcl/Tk due to its potential to cooperate well in every Xwindows environment and in different operational systems (irix, unix, solaris, etc.). We have also finished the interface for Linux, Unix, Windows NT and Windows 2000, WindowsXP. Most of our effort in the next few months is to check the validity of our codes in Linux and different Unix environments in order that it be used by any user in any operational system.

As regards the development of the beam-loading code with azimuthal dependence of eigenmodes of the beam tunnel, we have written the analysis and some parts of the numerical code for a non-periodic corrugated waveguide, in which the eigenvalue problem is solved for complex dielectric constant. This is an important extension of our previous codes, in which we solve the system of equations taking into account only the real part of the dielectric constant, i.e., we look for real roots of the determinant of the truncated system of equations. In this subtask, at first we will solve the complex determinant. For this purpose we have transformed in C an algorithm provided by Prof. Patrick Queffelec ["Nonreciprocal cell for the broad-band measurement of tensorial permeability of magnetized ferrites: direct problem", IEEE Trans. Microwave Theory Techn. 47, 390-397 (1999)]. We have made the necessary test runs in order to understand the procedure in which that subroutine finds the complex roots of a complex function. In addition, Dr. A. Manenkov from the Russian Academy of Sciences, who visited University of Athens for cooperation, has provided us one more subroutine, which calculates complex roots of a complex function. Now we are testing this subroutine and comparing its speed to that of Prof. Queffelec. [It should be noted that in our case the determinant is not an analytic function. Therefore, special attention has to be given to the use of these subroutines. Note also that in the case of complex relative dielectric permittivity the analysis for the electromagnetic energy and the quality factor in the beam tunnel has to be made again, because we have to introduce in the expressions of the fields the losses caused by the imaginary part of dielectric permittivity.]

Finally, preliminary attention was given to the question of the analysis of a corrugated waveguide with a step between the metallic and dielectric parts. This subtask has been added during this year. In particular, we have modified our analysis in order to calculate the dispersion relation and the electromagnetic field components of a periodic corrugated waveguide, in which the inner radius of the metallic walls do not coincide with the inner radius of the dielectric material. Up to now we have finished the analytical work and during 2002 we expect to finish the respective numerical code in C for all kinds of modes.

### 2a4) Coaxial and harmonic gyrotrons

(*This activity was performed** under the cost-sharing contract ERB 5004 CT98 0019, assumed before the Contract of Association was signed. The prescribed work has been completed and the Final Report accepted by the Commission. Relevant summary information is presented in ***Annex IV**.)

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

During the period in subject, work was confined to studying the theoretical aspects of the ASDEX-Upgrade SOL and divertor plasma behavior. In particular:

We have mostly worked on the gyrofluid equations and the possibility of deriving them under a rigid Lagrangian formalism. Following the basic method of *Pfirch* *and Correa-Restrepo (IPP report, 2000), *which was made for a drift kinetic treatment of the plasma, we have derived the gyrofluid Lagrangian straight from the gyrokinetic, gyroangle independent Lagrangian *(Hahm, 1998).* From it we have derived the momentum equation of the gyrofluid, keeping the drifts in the final equation. Using the density conservation law derived in *Dorland and Hammett (Phys. Fluids B, 1993)*, and ignoring the drifts apart from the **ExB** drift, we were able to obtain the same results as those obtained by *Dorland and Hammett*, where the equations had been derived from the Vlasov equation.

To have another proof of the validity of our method, we have also compared our results to those obtained from *Pfirsh and Correa-Restrepo* in the drift kinetic regime. We have thus proved an equality between the drift kinetic and the gyrokinetic Lagrangian, in the limit when the electric potential is arbitrary and the perpendicular scale length becomes small. In that limit, we have found that our results and those of *Pfirsh and Correa-Restrepo *are matching. This result is interesting, since we can now use in this limit either of the two Lagrangians, originally derived for different purposes in describing the plasma.

From the gyrofluid Lagrangian we built, we also found a law for the conservation of energy, by using Noether’s theorem. In this way, the energy conservation law becomes exact (since no higher order moments are cut off by deriving it) and self- consistent (since it is derived out of the basic laws of mechanics).

The derivation of the self-consistent gyrofluid equations, together with the energy conservation theorem, is important for the computational codes for gyrofluid. Thus this work will help in building a self-consistent code, which will also contain the energy conservation and will study the anomalous perpendicular transport in the edge. This will help many parts in plasma physics, as drift-wave turbulence and its associated anomalous transport, description of the SOL, etc.

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

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

The period in subject was devoted to mainly preparatory work on this activity.

### 2c2) MHD turbulent transport in plasmas

During the period in subject, the work in this field was mainly focused in extending and using the CFD codes (FUSION3D, FUSION2D) developed in the Laboratory for MHD flows. These source codes are basically Navier-Stokes solvers using the finite volume approach. The FUSION3D code (based on the DIAN3D code) can simulate steady or time-dependent laminar and eventually turbulent MHD flows in cartesian or cylindrical coordinates. The FUSION2D code (based on the TEACH code) can simulate steady laminar MHD flows, in particular natural convection. The following MHD flows have been investigated:

1. Developing Hartmann flows: These case studies were mainly undertaken in order to test the validity of the above codes (see Annex V)

2. MHD natural convection with Low-Rm approximation: The low-Rm approximation is currently used by several researchers because of its lower computational effort. MHD flows in rectangular enclosures subject to transverse homogeneous magnetic fields have been studied in order to test the code and compare with recent research works (see Annex VI)

3. Validity of the Low-Rm approximation vs. full induction calculations: MHD flows in rectangular enclosures have been studied in order to establish the range of the validity of the low-Rm approximation (see Annex VII)

4. Effect of magnetic field direction on MHD natural convection: MHD flows in rectangular enclosures have been studied in order to assess the effect of the direction of the magnetic field on the flow and heat transfer (see Annex VIII)

5. MHD natural convection with internal heat sources: The FUSION2D code has been extended to include MHD flows in rectangular enclosures with internal heat sources (see Annex IX)

The collected computational results appear in: a) *I.E. Sarris, A. Grecos **and N.S. Vlachos, “Development of CFD models turbulent plasma flows**” (**in** **Greek**), 2*^{nd}* Meeting on Research Activities in Fluid Mechanics in Greece, Univ. of Thessaly, Volos, Greece, May 25, 2000, *b)* G**. **Zikos**, “**Study of the effects of magnetic fields on boundary layer and natural convection flows**” (**in** **Greek**), **Diploma** **Thesis**, **University** **of** **Thessaly**, Oct. 2001*, c) *S. Kakarantzas, ‘Modeling of Natural Convection of **Liquid Metals with Internal Heat Sources under the Influence of Magnetic Boundary Layer’, Diploma** **Thesis**, **Univ.** **of** **Thessaly, in progress, *and d) *A. Tataridou, ‘Development of Computational Model for Turbulent Flows in Plasmas‘, Master** **Thesis**, **Univ.** **of** **Thessaly, in progress.*

The extension of the codes to include cylindrical geometries and turbulence effects is being investigated. However, the work of this period has mainly focused on MHD natural convection, a domain where research is very active because of its importance in heat transfer problems. The codes developed by our research group can handle efficiently this kind of flows.

### 2c3) Stochastic modelling of transport phenomena

During the period in subject, the kinetic equations for charged test particles have been addressed. This work concerns the derivation and numerical investigation of kinetic equations, mainly of Fokker-Planck type, governing the evolution of the probability density of a charged test particle. A homogeneous external magnetic field acts on the particles as well as a random force due to the (weak) interaction with the bulk of a plasma. The Landau equation is generalised by taking into account, consistently, the effect of the magnetic field in the collision term. Furthermore, the properties of the modified kinetic equation are studied and compared with other equations proposed in the literature. In particular, techniques of non-equilibrium stability mechanics have been applied leading to an equation where the effect of the motion of the particle perpendicular to the field lines to the collision term is correctly accounted. However, problems arise when considering the motion along the field lines, a difficulty encountered in the theory of open systems. To clarify this question, the case of a simple one-dimensional model with ‘coloured’ noise as the random force has been studied. Work in this direction has been continued to cover the three-dimensional case of a magnetised test particle in thermal equilibrium.

Also, during the same period, expressions for various transport coefficients have been obtained. Taking into account the axial symmetry of the system, these coefficients have been put in a form suitable for numerical calculations.

As regards the lattice Boltzmann models for MHD flows with boundary conditions, analytical solutions for the Hartmann flow have been derived for the 13-bit hexagonal model and comparison with the analytical profiles have been performed. A first order error has been detected indicating the need of deriving LB-MHD models with superior numerical stability properties. Numerical codes have been developed and tests have been performed. There is good agreement between the results and the analytical expressions. The code remains stable for small values of the Reynolds number and the pressure gradient (*see ***Annex X**). The collected results appear in: *D. Valougeorgis and S. Naris, "An analytical Lattice Boltzmann solution for thermal flow problems", 17th International Conference on Transport Theory, Imperial College, London, England, (2001).*

A parallel activity during the same period aims at understanding the kinetics and energetics of particles in a turbulent environment with fractally distributed electric fields. This investigation includes the particular case where the turbulent plasma is modeled by Self-Organized Criticality (SOC), which is characterized by a unique fractal dimension. In particular (**Annex XI**), we analyzed the (constant velocity) random walk of particles through a fractal environment, where the particles are scattered off into random directions when they meet a field inhomogeneity. We analytically derived the probability distribution *p*_{r} of the random walk increments of particles in between two subsequent encounters with field inhomogeneities. It turns out that* p*_{r }depends basically just on the dimension D of the fractal. For D<2, *p*_{r} is quite well approximated by a power-law, the second moment is infinite, and the random walk is of the Levy type. We analyzed the diffusive behaviour of the particles in the frame of Continuous Time Random Walks, and we showed that the particles undergo anomalous, enhanced diffusion for D<2. For D>2, diffusion is normal. In particular, we showed that SOC gives rise to enhanced diffusion. We confirmed our results with Monte-Carlo simulations.* (This activity will be continued into the following period.** The results are presented in a paper submitted to Phys. Rev. Lett.: Random walk through fractal environments: An example of Levy walks, H. Isliker, L. Vlahos, 2001.**)*

### 2c4) Assessment of the theory of non-resonant stabilisation by superthermal particles

The theory of the non-resonant suppression of MHD modes (such as the sawtooth instability and the neoclassical tearing modes) by superthermal particles (i.e. ions and electrons) during r.f. heating (i.e. ICRH and ECRH) was recently developed and published [*A. Lazaros,** **Physics of Plasmas, Vol.6, pp.148-152 (1999);** **Physics of Plasmas, Vol.8, pp.1263-1266 (2001); Fusion Engineering and Design, Vol.53, pp.35-42 (2001) *and* **Physics of Plasmas, Vol.8, pp.3695-3701 (2001)*] During the period in subject, preliminary experimental evidence for these effects was obtained, and discussions for dedicated experiments in TEXTOR and Tore-Supra have been conducted, during the mobility of A.Lazaros at FZ-Juelich (May 2001) and CEA-Cadarache (November 2001) respectively.

During the collaboration of A.Lazaros with the TEXTOR group (and in particular with E.Westerhof), cross-island diffusion of superthermal ions (which is a major issue for the non-resonant interaction) has been resolved and a chapter on this matter was included in the relevant publication. In the analysis of the existing data from TEXTOR (before the shut-down) no conclusive evidence could be found, regarding superthermal production during ECRH, because of the very low ECRH power of 0.2MW at these experiments. The required ECRH power for a detectable growth of the superthermal population was evaluated and it was confirmed that the available power of 1MW (after the shut-down) would be sufficient. We have been also discussing methods to monitor the superthermal emission and we agreed to do so by filtering out the thermal part from the soft-X-ray intensity. Dedicated experiments on the suppression of neoclassical tearing modes were proposed for the new experimental campaign from January 2003.

During the collaboration of A.Lazaros with the Tore-Supra Group (and in particular with M.Ottaviani, F.Nguyen, G.Huysmans, L.Colas, P.Maget) an overview of all experimental results, regarding neoclassical tearing modes and sawteeth stability during ICRH, was conducted and comparisons with the JET results were made. In these results very clear evidence of the stabilising effect of the superthermal ions was found. The most remarkable result, however, of this collaboration has been the discovery (or more precisely the identification), for the first time, of neoclassical tearing modes in the magnetic fluctuations in Tore-Supra, although the data were collected three years ago. The findings of the study were included in a CEA report (**Annex XV**) entitled “*Overview of Sawteeth Stabilization and Correlation with NTMs during ICRH in Tore-Supra*”. The experimental program for further investigations on NTM and sawtooth stability was prepared and proposed for the experimental campaign of 2003 with ICRH.

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

Continuing this activity into the present period, stable regions of the Mathieu equation were determined by applying a sufficient condition for the linear stability of generic nonoautonomus stability dissipative mechanical systems [*H. Tasso, G. N. Throumoulopoulos, Phys. Lett. A. ***271***, 413 (2000**)*]. The full stability charts of the Mathieu and Hill equations associated with necessary and sufficient conditions were also obtained. They show sizable stable regions for the negative-energy mode due to combined action of dissipation and parametric excitation. By analogy, these results are a strong indication that the resistive wall mode could be stabilized by the joint action of a properly tailored time-dependent wall resistivity and a sufficient viscous dissipation in the plasma (**Annex**** XII**). In addition, cylindrical plasma equilibria with reversed magnetic shear and sheared flow in connection with internal transport barriers in tokamaks have been investigated (**Annex X****III**), while a study on toroidal axisymmetric equilibria with flow and anisotropic conductivity is briefly reported in **Annex X****IV**. Finally, the equilibrium equations of a toroidal axisymmetric plasma with compressible flow and isothermal magnetic surfaces were derived; they consist of a differential equation for the poloidal magnetic flux function coupled through the density with an algebraic Bernoulli equation. Due to that coupling the above mentioned set of equations should, in general, be solved numerically.

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

During the present period, the modules of the time dependent quasi-three-dimensional code have been coupled to the nonlinear 3-D MHD code M3D to study the expansion dynamics of pellet clouds in truly 3D magnetic fields. The quasi-three-dimensional pellet code calculates all major characteristics of the pellet ablation process. The M3D code is a multilevel MHD code with an unstructured triangular grid in the poloidal plane and prism elements in the 3D space. The coupling of the two codes is as follows: a pellet is injected into a torus with certain velocity along a prescribed flight coupling of the two codes path. As the pellet traverses the torus (simulated by the M3D code) the pellet resides along its path at discrete positions for certain residence times. During these residence times M3D provides the parameters for the background magnetized plasma and the ablation code is used to calculate the formation of a confined plasmoid around the ablating pellet. These plasmoids, aligned with the magnetic field lines, are then transferred to the M3D code that computes the further evolution of these local massive disturbances. For our studies of pellet-plasma interaction in real tokamaks(e.g. Asdex Upgrade) we interfaced the DIVA equilibrium code to M3D. The initial conditions of the M3D code prior to the pellet injection (plasma geometry, the numerical grid, and various other plasma parameters) are taken from the DIVA code. [More details are presented in **Annex XVII**. Some preliminary results were presented to the EPS Conference in Madeira.]

In addition (**Annex XVI**), the Sloan profile and wave front reduction algorithm has been applied to the 2D vapour shield code for node renumbering. No extensive testing of the renumbered numerical grid has been made yet, since the effort has concentrated in modifying the 2-D vapor shield code into a 2-D pellet ablation code.

Last Updated (Friday, 04 February 2011 16:16)