EC Waves

Electron Cyclotron Waves

Electron cyclotron (EC) waves are electromagnetic oscillations propagating in magnetized plasma at frequencies around the electron cyclotron frequency and its harmonics. For current Tokamaks, with a magnetic field strength in the range of 3 to 6 T, this frequency lies in the 100 to 200 GHz range. In this frequency range waves of mm-wavelength propagate freely in vacuum. This is an important aspect of EC waves in their application for heating and current drive of Tokamak plasmas; no physical contact between the wave launching antenna and the plasma is required thus avoiding plasma power loading, erosion and contamination, and, in addition, no impedance matching problems exist.

Resonant interaction with the electrons occurs where the wave frequency coincides with the local EC frequency or its harmonics. The mm-wave beam can be focused down to a size of a few cm across. Therefore, this interaction and the concurrent power absorption becomes extremely localized, which makes EC waves ideally suited for localized plasma heating and current drive. In particular, the property of current drive to control magneto-hydrodynamic (MHD) instabilities is of prime importance to ITER. The activities of the Computational Plasma Physics High Temperature Group cover a broad range of linear and nonlinear plasma wave propagation and absorption, including wave transport from launcher to the plasma core as well as launcher design calculations.

Modelling

Models for plasma wave propagation and absorption have been developed at various levels of physical complexity. Linear wave propagation models include simple ray-tracing, Gaussian beam tracing including effects of diffraction, as well as quasi-optical models for more complicated beam behaviour. The nonlinear plasma response is commonly modelled in terms of the quasi-linear wave diffusion as embodied by the bounce averaged, quasi-linear Fokker-Planck equation. These models are incorporated in a number of numerical modelling codes. Modelling supports the FOM ECRH and ECCD experiments on TEXTOR and AUG. Also, the ITER ECRH system design is supported by comprehensive modelling of the ITER ECRH launcher design options under various ITER scenarios.

TORAY ray-tracing code

The TORAY ray-tracing code (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 Gaussian wave beam by an appropriate set of individual rays. The code solves the Hamiltonian ray-tracing equations in the geometry of general Tokamak equilibrium. 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 the adjoint calculation as incorporated in the CURBA set of routines (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 small 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) (A.A. Balakin, et al., Nucl. Fusion 48 (2008) 065003 and N. Bertelli, et al., Nucl. Fusion 50 (2010) 10). These calculations reveal a quite complicated behaviour of the wave beam in the electron cyclotron resonance layer, as illustrated by the Figure. This behaviour is a consequence of the inhomogeneity in the dispersion of the EC resonance region at beam width scale both in propagation and absorption of the waves. The aberrations also have a profound influence on the achievable width of the power deposition profile which is important for localised current drive on the scale of the NTM magnetic island width. In comparison to the usual beam tracing predictions, power deposition profiles are found to be broadened by 15 to 30%.
 

The main task of the ITER ECRH UPL is the control of neoclassical tearing (NTM) modes and saw-teeth oscillations. The effectiveness of the injected EC wave power for both of these applications strongly depends on the power deposition profile. The broader the deposition profile, the more power is needed for ITER saw-tooth and NTM control. ECCD efficiency has been calculated by the quasi-optical code employing the adjoint method. Efficiencies are found to be reduced by up to 20%, which is of serious consequence to the definition and the deployment of the ITER ECRH system. 
 

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