O. Maj

Simulations of combined neutral beam injection and ion cyclotron heating with the TORIC-SSFPQL package

A source describing the injection of fast ions due to the ionization of high-energy neutral beams has been added to the surface-averaged quasilinear Fokker–Planck code SSFPQL (Brambilla 1994 Nucl. Fusion 34 1121). For this purpose, the multiple-beam NBI code SINBAD (Feng et al 1995 Comput. Phys. Commun. 88 161) has been included as a module in SSFPQL, with the modifications required to handle arbitrary axisymmetric equilibria. Alternatively, the neutral beam injection (NBI) source can be built using the output of a Monte Carlo NBI code. We have also added a term describing losses of fast ions during thermalization, and a subroutine evaluating the neutron production rate by nuclear reactions. With these extensions, iterations between SSFPQL and the full-wave solver TORIC (Brambilla 1999 Plasma Phys. Control. Fusion 41 1) can now be used to investigate the strong interplay between NBI and ion cyclotron (IC) heating.By comparing the predicted and measured neutron production rates from D–D reactions in a discharge with combined NBI and IC heating in ASDEX Upgrade we obtain a plausible estimate of the importance of fast-ion losses (FILs), even if their cause cannot be identified. We find, however, that the plasma composition, in particular the presence of low Z impurities, plays a more critical role than FILs in limiting the efficiency of this heating scheme.

On recent results in the modelling of neoclassical-tearing-mode stabilization via electron cyclotron current drive and their impact on the design of the upper EC launcher for ITER

Electron cyclotron wave beams injected from a launcher placed in the upper part of the vessel will be used in ITER to control MHD instabilities, in particular neoclassical tearing modes (NTMs). Simplified NTM stabilization criteria have been used in the past to guide the optimization of the launcher. Their derivation is reviewed in this paper and their range of applicability clarified. Moreover, possible effects leading to a deterioration of the predicted performance are discussed. Particularly critical in this context is the broadening of the electron-cyclotron (EC) deposition profiles. It is argued that the most detrimental effect for ITER is likely to be the scattering of the EC beams from density fluctuations due to plasma turbulence, resulting in a beam broadening by about a factor of two. The combined impact of these effects with that of beam misalignment (with respect to the targeted surface) is investigated by solving the Rutherford equation in a form that retains the most relevant terms. The perspectives for NTM stabilization in the Q = 10 ITER scenario are discussed.

Effects of aberration on paraxial wave beams: beam tracing versus quasi-optical solutions

This paper aims to clarify the role of aberration effects on the propagation and absorption of wave beams in inhomogeneous dispersive and dissipative media. We consider models in which aberration effects can be caused by the presence of either caustics or spatially dispersive absorption, with reference to the propagation near a cut-off or to the electron-cyclotron (EC) resonance, respectively. For such models, the standard beam tracing description of paraxial wave beams and the recently proposed quasi-optical method, which accounts for aberration, are compared and verified on the basis of the analytical exact solutions. We find that the presence of a cut-off implies no significant aberration of the beam, while significant aberration is found when dispersive absorption is so strong that different wavenumbers in the beam spectrum are damped at different locations. This phenomenon is well described by the quasi-optical method. Finally, an extrapolation of this simple two-dimensional model to the case of the ITER upper EC port is addressed with the result that the broadening of the power deposition profiles never exceeds 10%.

The field line map approach for simulations of magnetically confined plasmas

In the presented field line map approach the simulation domain of a tokamak is covered with a cylindrical grid, which is Cartesian within poloidal planes. Standard finite-difference methods can be used for the discretisation of perpendicular (w.r.t.~magnetic field lines) operators. The characteristic flute mode property

Validation of the paraxial beam-tracing method in critical cases

\left(k_{\parallel}\ll k_{\perp}\right)

Paraxial Wentzel–Kramers–Brillouin method applied to the lower hybrid wave propagation

of structures is exploited computationally by a grid sparsification in the toroidal direction. A field line following discretisation of parallel operators is then required, which is achieved via a finite difference along magnetic field lines. This includes field line tracing and interpolation or integration. The main emphasis of this paper is on the discretisation of the parallel diffusion operator. Based on the support operator method a scheme is constructed which exhibits only very low numerical perpendicular diffusion. The schemes are implemented in the new code GRILLIX, and extensive benchmarks are presented which show the validity of the approach in general and GRILLIX in particular. The main advantage of the approach is that it does not rely on field/flux-aligned, which become singular on the separatrix/X-point. Most tokamaks are based on the divertor concept, and the numerical treatment of the separatrix is therefore of importance for simulations of the edge and scrape-off layer.

An algorithm for fast Hilbert transform of real functions

Paraxial Wentzel–Kramers–Brillouin (WKB) (beam-tracing) solutions of a scalar wave equation are compared to the exact solutions for the two important cases of reflection from a plasma with linearly increasing density and of oblique propagation in the presence of an absorbing half plane. In these situations, the beam-tracing technique is close to its limits of validity, yet these conditions are often met in practical applications. More specifically, the first case is relevant to reflectometry diagnostics, whereas the second case models the absorption of electron-cyclotron beams obliquely launched onto the resonance layer, as envisaged, for instance, for the ITER upper launcher [M. A. Henderson et al., Nucl. Fusion 48, 054013 (2008)]. In both cases, the beam-tracing approach is found to reproduce well the exact behavior of the wave for experimentally relevant parameters, confirming the robustness of WKB-based techniques close to or even beyond their range of applicability. For the latter case, moreover, the...

Dynamical coupling between magnetic equilibrium and transport in tokamak scenario modelling, with application to current ramps

The paraxial Wentzel–Kramers–Brillouin (pWKB) approximation, also called beam tracing method, has been employed in order to study the propagation of lower hybrid waves in a tokamak plasma. Analogous to the well-know ray tracing method, this approach reduces Maxwell’s equations to a set of ordinary differential equations, while, in addition, retains the effects of the finite beam cross-section, and, thus, the effects of diffraction. A new code, LHBEAM (lower hybrid BEAM tracing), is presented, which solves the pWKB equations in tokamak geometry for arbitrary launching conditions and for analytic and experimental plasma equilibria. In addition, LHBEAM includes linear electron Landau damping for the evaluation of the absorbed power density and the reconstruction of the wave electric field in both the physical and Fourier space. Illustrative LHBEAM calculations are presented along with a comparison with the ray tracing code GENRAY and the full wave solver TORIC-LH.

A stable scheme for computation of coupled transport and equilibrium equations in tokamaks

A simple and accurate algorithm to evaluate the Hilbert transform of a real function is proposed using interpolations with piecewise–linear functions. An appropriate matrix representation reduces the complexity of this algorithm to the complexity of matrix-vector multiplication. Since the core matrix is an antisymmetric Toeplitz matrix, the discrete trigonometric transform can be exploited to calculate the matrix–vector multiplication with a reduction of the complexity to O(N log N), with N being the dimension of the core matrix. This algorithm has been originally envisaged for self-consistent simulations of radio-frequency wave propagation and absorption in fusion plasmas.

Modelling the influence of temperature anisotropies on poloidal asymmetries of density in the core of rotating plasmas

The modelling of tokamak scenarios requires the simultaneous solution of both the time evolution of the plasma kinetic profiles and of the magnetic equilibrium. Their dynamical coupling involves additional complications, which are not present when the two physical problems are solved separately. Difficulties arise in maintaining consistency in the time evolution among quantities which appear in both the transport and the Grad–Shafranov equations, specifically the poloidal and toroidal magnetic fluxes as a function of each other and of the geometry. The required consistency can be obtained by means of iteration cycles, which are performed outside the equilibrium code and which can have different convergence properties depending on the chosen numerical scheme. When these external iterations are performed, the stability of the coupled system becomes a concern. In contrast, if these iterations are not performed, the coupled system is numerically stable, but can become physically inconsistent. By employing a novel scheme (Fable E et al 2012 Nucl. Fusion submitted), which ensures stability and physical consistency among the same quantities that appear in both the transport and magnetic equilibrium equations, a newly developed version of the ASTRA transport code (Pereverzev G V et al 1991 IPP Report 5/42), which is coupled to the SPIDER equilibrium code (Ivanov A A et al 2005 32nd EPS Conf. on Plasma Physics (Tarragona, 27 June–1 July) vol 29C (ECA) P-5.063), in both prescribed- and free-boundary modes is presented here for the first time. The ASTRA–SPIDER coupled system is then applied to the specific study of the modelling of controlled current ramp-up in ASDEX Upgrade discharges.