W. Kernbichler

Evaluation of 1/ν neoclassical transport in stellarators

Using an analytic solution of the kinetic equation in the 1/ν regime, a new formula for the neoclassical transport coefficients is obtained which takes into account all classes of trapped particles. This formula holds in any coordinate system and no simplifying assumptions about the magnetic field are needed. Therefore it is also applicable to complex magnetic fields given in real space coordinates. The method and the results can be used to optimize magnetic field configurations with respect to the 1/ν regime. The method is bench-marked against Monte Carlo calculation both for the l=3 classical stellarator model and also for the original Helias configuration [J. Nuhrenberg and R. Zille, Phys. Lett. A 114, 129 (1986)] with a more complex magnetic field structure. Some features of transport for Helias are clarified by analyzing the bounce-averaged drift velocity.

Kinetic estimate of the shielding of resonant magnetic field perturbations by the plasma in DIII-D

Effects of linear plasma response currents on non-axisymmetric magnetic field perturbations from the I-coil used for edge localized mode mitigation in DIII-D tokamak are analysed with the help of a kinetic plasma response model developed for cylindrical geometry. It is shown that these currents eliminate the ergodization of the magnetic field in the core plasma and reduce the size of the ergodic layer at the edge. A simple balance model is proposed which qualitatively reproduces the evolution of the plasma parameters in the pedestal region with the onset of the perturbation. It is suggested that the experimentally observed density pump-out effect in the long mean free path regime is the result of a combined action of ion orbit losses and magnetic field ergodization at the edge.

Benchmarking of the mono-energetic transport coefficients—results from the International Collaboration on Neoclassical Transport in Stellarators (ICNTS)

Numerical results for the three mono-energetic transport coefficients required for a complete neoclassical description of stellarator plasmas have been benchmarked within an international collaboration. These transport coefficients are flux-surface-averaged moments of solutions to the linearized drift kinetic equation which have been determined using field-line-integration techniques, Monte Carlo simulations, a variational method employing Fourier–Legendre test functions and a finite-difference scheme. The benchmarking has been successfully carried out for past, present and future devices which represent different optimization strategies within the extensive configuration space available to stellarators. A qualitative comparison of the results with theoretical expectations for simple model fields is provided. The behaviour of the results for the mono-energetic radial and parallel transport coefficients can be largely understood from such theoretical considerations but the mono-energetic bootstrap current coefficient exhibits characteristics which have not been predicted.

Monte Carlo study of heat conductivity in stochastic boundaries: Application to the TEXTOR ergodic divertor

The heat balance equation is derived and solved for fusion edge plasma conditions with (partially developed) ergodic magnetic-field structures. For this purpose, a three-dimensional (3D) Monte Carlo code, “E3D,” based upon the “multiple local magnetic coordinate system approach” has been developed. Parameters typical for the Dynamic Ergodic Divertor (DED) of TEXTOR-94 (Torus Experiment for the Technology Oriented Research) [K. H. Finken et al., Fusion Eng. Des. 37, 1 (1997)] are chosen in the applications. The plasma temperature fields and the profiles of the radial component of heat flux due to the classical parallel and anomalous perpendicular diffusion are calculated. Because of magnetic-field ergodization and diversion of field lines, parallel conduction also can contribute to this radial flux. The results are compared with theoretical predictions for two limiting cases: With the Rechester–Rosenbluth model of ergodization-induced transport and with a “laminar flow model” proposed in the present paper. This latter model describes the effects of field line diversion. The diversion effect is shown to be dominant for TEXTOR-DED conditions.

Evaluation of the toroidal torque driven by external non-resonant non-axisymmetric magnetic field perturbations in a tokamak

The toroidal torque driven by external non-resonant magnetic perturbations (neoclassical toroidal viscosity) is an important momentum source affecting the toroidal plasma rotation in tokamaks. The well-known force-flux relation directly links this torque to the non-ambipolar neoclassical particle fluxes arising due to the violation of the toroidal symmetry of the magnetic field. Here, a quasilinear approach for the numerical computation of these fluxes is described, which reduces the dimension of a standard neoclassical transport problem by one without model simplifications of the linearized drift kinetic equation. The only limiting condition is that the non-axisymmetric perturbation field is small enough such that the effect of the perturbation field on particle motion within the flux surface is negligible. Therefore, in addition to most of the transport regimes described by the banana (bounce averaged) kinetic equation also such regimes as, e.g., ripple-plateau and resonant diffusion regimes are naturally included in this approach. Based on this approach, a quasilinear version of the code NEO-2 [W. Kernbichler et al., Plasma Fusion Res. 3, S1061 (2008).] has been developed and benchmarked against a few analytical and numerical models. Results from NEO-2 stay in good agreement with results from these models in their pertinent range of validity.

Calculation of the bootstrap current profile for the TJ-II stellarator

Calculations of the bootstrap current for the TJ-II stellarator are presented. DKES and NEO-MC codes are employed; the latter has allowed, for the first time, the precise computation of the bootstrap transport coefficient in the long mean free path regime of this device. The low error bars allow a precise convolution of the monoenergetic coefficients, which is confirmed by error analysis. The radial profile of the bootstrap current is presented for the first time for the 100_44_64 configuration of TJ-II for three different collisionality regimes. The bootstrap coefficient is then compared to that of other configurations of TJ-II regularly operated. The results show qualitative agreement with toroidal current measurements; precise comparison with real discharges is ongoing.

Electron cyclotron current drive in low collisionality limit: On parallel momentum conservation

A comprehensive treatment of the models used in ray- and beam-tracing codes to calculate the electron cyclotron current drive (ECCD) by means of the adjoint technique, based on the adjoint properties of the collision and Vlasov operators appearing in the drift-kinetic equation, is presented. Particular attention is focused on carefully solving the adjoint drift-kinetic equation (generalized Spitzer problem) with parallel momentum conservation in the like-particle collisions. The formulation of the problem is valid for an arbitrary magnetic configuration. Only the limit of low collisionality is considered here, which is of relevance for high-temperature plasmas. It is shown that the accurate solution of the adjoint drift-kinetic equation with parallel momentum conservation significantly differs (apart from the suprathermal electron portion) from that calculated in the high-speed-limit, which is most commonly used in the literature. For high-temperature plasmas with significant relativistic effects, the accuracy of the resulting numerical models is demonstrated by ray-tracing calculations and benchmark results are presented. It is found that the ECCD efficiency calculated for ITER with parallel momentum conservation significantly exceeds the predictions obtained with the high-speed-limit model.

Kinetic modelling of the interaction of rotating magnetic fields with a radially inhomogeneous plasma

The interaction of rotating magnetic fields (RMFs) with a plasma is modelled in the linear approximation. A kinetic Hamiltonian model for the rf plasma conductivity is used. A radially inhomogeneous periodic cylindrical plasma with a rotational transform of the magnetic field is studied with parameters relevant to the dynamic ergodic divertor (DED) of TEXTOR. For the case of a finite electron diamagnetic velocity it is shown that the torque resulting from the RMF tends to bring the electron fluid approximately to the rest frame of this field. This result is in qualitative agreement with long mean-free path drift MHD theory. In contrast to that theory where a resonant behaviour is found at electron and ion diamagnetic frequencies, in the present kinetic analysis, the RMF frequency where the torque passes through zero is smaller than the electron diamagnetic frequency if there is an electron temperature gradient present. The relation of these results with recent experimental measurements of the DED-induced plasma rotation in TEXTOR is discussed.

Variance reduction in computations of neoclassical transport in stellarators using a δf method

An improved δf Monte Carlo method for the computation of neoclassical transport coefficients in stellarators is presented. Compared to the standard δf method without filtering, the computing time needed for the same statistical error decreases by a factor proportional to the mean free path to the power 3∕2.

Mapping technique for stellarators

The Stochastic Mapping Technique (SMT), a highly efficient method to solve the five-dimensional drift kinetic equation in the long-mean-free-path regime, is presented in an application to stellarators. Within this method, the dimensionality of the problem is reduced to four dimensions through a discretization in one dimension. Instead of tracing test particles in the whole phase space, test particles are followed on particular Poincare cuts. With this approach, the computation time is reduced by a large factor compared to direct Monte Carlo methods. The SMT is applicable to stellarators with arbitrary magnetic field geometries and topologies. It can be used for any problem where currently conventional Monte Carlo methods are applied. In particular, it is well suited for modeling the distribution function of supra-thermal particles generated by auxiliary heating methods, for studies of stellarator transport properties and for a fast survey of a specific configuration in the whole phase space necessary for ...