EC Waves

Electron Cyclotron Waves

Electron cyclotron (EC) waves are electromagnetic oscillations in magnetized plasmas which are characterized by frequencies in the range of the electron cyclotron frequency or its harmonics. The work of the group is concerned with the behavior and usage of EC waves in high temperature fusion plasmas. Typical frequencies are in the 100 GHz range. In this frequency range, the microwaves generally propagate freely from vacuum to well inside the plasma. Resonant interaction with the electrons occurs in the region where the wave frequency coincides with the EC frequency or a harmonic thereof. Because microwave beams in this frequency range can be easily be focused down to a beam waist of a few cm this interaction and the consequent power absorption or wave emission is extremely localized. This makes EC waves ideally suited for very localized plasma heating or high resolution plasma diagnostics. The work of the group within this topic then also is focused on those applications making use of this special property of EC waves: in particular, the use of high power microwaves for the control of magnetohydrodynamic (MHD) instabilities by localized resonant heating (ECRH) or current drive (ECCD). The activities of the group cover a broad scope within linear and nonlinear wave theory, modelling of propagation, as well as experiments.

Theory

The major interest is in possible nonlinear aspects of wave propagation. Parametrically induced transparency has been proposed as a possible diagnostic for overdense plasma. The possible origin of strong perturbations, as observed on the in-line ECE signal during simultaneous ECRH (link to in-line ECE sub-subsection), from nonlinear instabilities in high power EC wave propagation is investigated.

Parametrically Induced Transparency

Parametrically induced transparency (PIT) is a variant of electromagnetically induced transparency (EIT). The PIT scheme is proposed as a diagnostic of overdense plasma and consists of the following steps: a) propagation of spontaneous plasma emission (signal wave) from the UHR region and it’s transformation to a so-called "transport" wave, excited by beating between signal and a high power drive wave; b) propagation of the "transport" wave to vacuum through the region, which is opaque for the signal wave but transparent for the transport wave; c) the signal wave intensity can be determined via appropriate post-processing of the received transporting wave.

The scenario for a proof-of-principle PIT experiment at the TEXTOR tokamak has been developed.

Figure 1: The figure illustrates the principle of parametrically induced transparency: ECE or Bernstein wave emission trapped inside the interior of an overdense plasma is transformed into a "transporting wave" by means of a nonlinear three wave interaction process between the waves themselves, a high power drive wave and the transporting wave. Optimal conditions for this transformation occur close to the upper-hybrid resonance of the trapped waves. The transporting wave can escape to vacuum carrying the information about the amplitude of the trapped waves.

Modelling

Models for wave propagation and absorption are created at various levels of physical complexity. Models for linear wave propagation include simple ray-tracing, beam-tracing for the description of Gaussian wave beams including effects of diffraction, as well as quasi-optical models for still more complicated beam behaviours. The nonlinear plasma response is commonly modelled in terms of the quasi-linear wave diffusion as embodied in the bounce averaged, quasi-linear Fokker-Planck equation. These models are incorporated in different numerical modelling codes.

Modelling is performed in support of the preparation and analysis of ECRH and ECCD experiments on TEXTOR. Also the ITER ECRH systems design is supported by comprehensive modelling of the effects of the high power EC waves in different ITER scenarios and for various ITER ECRH launcher design options.

TORAY ray-tracing code

The TORAY ray-tracing code (Reference A.H. Kritz et al.1982 Conf. Proc., 3rd Int. Symp. on Heating in Toroidal Plasmas ECE (Brussels, Belgium) vol 2 p 707; E. Westerhof, 1989, Rijnhuizen Report 89–183) models a Guassian wave beam by an appropriate set of individual rays. The code solves the Hamiltonian ray-tracing equations in the geometry of general tokamak equilibriums. The Hamiltonian is provided by either the cold or the warm plasma dispersion relation. Along each ray the power absorption is calculated using either a weakly or a fully relativistic calculation of the warm plasma dispersion. In addition an estimate of the non-inductively driven current is obtained from an adjoint calculation as incorporated in the CURBA set of routines (Reference: R.H. Cohen, Phys. Fluids 30 (1987) 2442).

TORBEAM beam-tracing code

The use of well focused beams is common in modern ECRH systems. The ray-tracing approximation breaks down near the focus of a wave beam where it predicts infinite power density. To improve on ray-tracing, beam-tracing models have bean developed which include effects of diffraction on the ray-trajectory or beam profile evolution. The TORBEAM code (Reference: E. Poli, et al., Comp. Phys. Commun. 136 (2001) 90) solves for the propagation of a Gaussian beam by solving for the trajectory of its central ray (given by the standard Hamiltonian ray-tracing equation) and additionally solving for the evolution of the beam width and phase front curvature in the plane perpendicular to the direction of beam propagation. Cold plasma dispersion is used to solve for the beam propagation. Absorption along the beam is calculated from the warm plasma dispersion relation (either weakly or fully relativistic) at the central ray. The CURBA set of routine is again used to estimate the non-inductively driven current. The model is limited to Gaussian beams and the assumption that the beam profile remains Gaussian.

Quasi-optical beam tracing

Various effects in the plasma may lead to non-Gaussian modifications of the beam profile. In particular, in the EC resonance region the relatively short scale spatial inhomogeneity and dispersion result in a break down of the assumptions underlying most beam-tracing models and the creation of non-Gaussian beam distortions, i.e. aberrations. Our collaborators of the Institute of Applied Physics in Nizhny Novgorod (Russia) have developed a quasi-optical model, which allows the description of wave beam evolution in the presence of aberrations (Reference: A.A. Balakin, et al., Nucl. Fusion 48 (2008) 065003). The model is based on a generalization of the parabolic wave equation of Fock and Leontovich (Reference: V.A. Fock, "Electromagnetic Diffraction and Propagation Problems" (1965) Oxford: Pergamon) and solves the evolution of the arbitrary beam profile in a plane perpendicular to a reference ray (close to but not necessarily identical to the beam centre). Absorption and beam propagation are solved self-consistently using the weakly relativistic warm plasma dispersion relations. Extension of the code to include an adjoint calculation of the current drive efficiency is envisaged.

RELAX bounce-averaged, quasi-linear Fokker-Planck code

The bounce-averaged quasi-linear Fokker-Planck equation describes the evolution of the particle distribution functions under the influence both collisions and waves. The equation is averaged over the bounce motion of the particles, and consequently describes the evolution of the flux surface averaged particle distribution functions in the low-collisionality, or banana regime. The RELAX code (Reference: E. Westerhof et al., 1992, Rijnhuizen Report RR92-211) solves the evolution of the electron distribution function under the influence of electron-electron as well as electron-ion collisions and the quasi-linear wave diffusion driven by the high power EC waves. The linearized, electron-electron collision operator can be modeled to different degrees of accuracy: high velocity limit, maxwellian or isotropic background, or a momentum conserving operator. Information on the EC wave beam properties for the calculation of the relativistic EC diffusion operator is to be transferred from the results of ray- or beam-tracing codes.

Highlight

Quasi-optical calculations have been performed to assess the performance of the ITER ECRH Upper Port front steering Launcher (UPL) (Reference: A.A. Balakin, et al., Nucl. Fusion 48 (2008) 065003). These calculations reveal a possibly quite complicated behaviour of the wave beam as illustrated in the figure. This behaviour is a consequence of the inhomogeneity of the propagation as well as the absorption and of the dispersion in the EC resonance region on the scale of the beam width. The aberrations also have a profound influence on the achievable width of the power deposition profile. In comparison to the usual beam tracing predictions, power deposition profiles are found to be broadened by up to a factor of 1.6.

The major task of the ITER ECRH UPL is the stabilization or control on neoclassical tearing modes and sawteeth. The effectiveness of the injected EC wave power for both these applications depends strongly on the power deposition width. The broader deposition profiles predicted by the quasi-optical code, would indicate that up to two times more power may be required for these tasks in ITER than hitherto considered. This has severe implications for the possible usages of the ITER ECRH systems.

Figure 2: Example of an EC wave beam trajectory in the ITER scenario 2 plasma equilibrium. The example shows a case with injection relatively oblique to the EC resonance layer. Under these conditions large aberrations are found: the centre of mass (red dashed curve) of the part of the beam "reflected" from the absorption layer no longer coincides with the trajectory of the central ray of the original beam.

Collaborations

• Institute of Applied Physics, Russian Academy of Sciences, Nizhny Novgorod, Russia: A.A. Balakin, M.A. Balakina (deceased), A.Yu. Kryachko, L.V. Lubyako, A.G. Shalashov, M.D. Tokman
• Max-Planck Institute for Plasma Physics, Garching, Germany: E. Poli
• Risø National Laboratory for Sustainable Energy, Danish Technical University, Roskilde, Denmark: H. Bindslev, S. Korsholm, S.K. Nielsen
• Russian Research Centre "Kurchatov Institute", Moscow, Russia: L.K. Kuznetsova
• CompX, DelMar (CA) USA: R.W. Harvey
• EFDA Taskforce on Integrated Tokamak Modelling