A. Bergmann

Two-dimensional particle simulation of Langmuir probe sheaths with oblique magnetic field

Two‐dimensional particle simulations of the sheath in front of a flat Langmuir probe mounted into a particle absorbing plate are performed to study the influence of a strong magnetic field (relation between Larmor radii and Debye length: ρe≤λD, ρi≫λD) which is oriented obliquely to the probe surface. Ion‐attracting probes are considered and the sheath is assumed to be collisionless. The full particle orbits in the homogeneous magnetic field and the self‐consistent electric potential are calculated with a particle‐in‐cell (PIC) code with two spatial coordinates and three velocity components (2d,3v). The main results are: In the sheath the ion trajectories are bent towards the normal to the probe surface so that the ion flow is focused to the edges of the probe. This leads to an enhancement of the ion current as compared to the current flowing in the flux tube subtended by the probe. As a consequence the current does not saturate at large (negative) probe voltage, because the thickness of the Debye sheath, ...

Transport of edge-localized mode energy in a scrape-off layer in the presence of collisionless fast electrons

The transport of the energy delivered by an edge-localized mode (ELM) into the scrape-off layer (SOL) of a tokamak plasma is studied with particle-in-cell simulations. A simple one-dimensional model with a Monte-Carlo procedure for Coulomb collisions is used. The SOL is collisional before the ELM, but the mean-free-path length in the hot ELM plasma is longer than the connection length, such that fast electrons can travel directly from the midplane to the target plate. However, the sheath potential increases rapidly, and most of the energy is transported with a velocity of order of the ion sound speed if secondary electron emission (SEE) is absent or weak. In the case of strong SEE, though, a large fraction of the energy can be carried by fast electrons during the ELM pulse.

Evidence for the neoclassical nature of the radial electric field in the edge transport barrier of ASDEX Upgrade

Experiments have been performed on ASDEX Upgrade to clarify the nature of the radial electric field, Er, in the edge transport barrier of tokamak plasmas. Highly resolved radial profiles of Er have been diagnosed spectroscopically through the radial force balance of impurity ions. We show that in the fully developed, highly collisional edge pedestal the nature of the radial electric field is neoclassical. This requires, in particular, that the main ion poloidal rotation is at neoclassical levels. Both main ion and impurity ion poloidal rotation profiles have been measured in deuterium, hydrogen and helium plasmas with main ion pedestal top collisionalities, ν*,i, between 1.2 and 12. These profiles have been compared to a hierarchy of neoclassical models (from a conventional description to a comprehensive model including finite orbit width effects). They are found, in all cases, to be in good agreement demonstrating that inside the edge transport barrier the Er well is sustained by the gradients of the main ion species.

Guiding center particle simulation of wide-orbit neoclassical transport

The neoclassical ion transport in a tokamak plasma with circular cross section is studied with guiding center particle simulations. A Monte Carlo model of pitch-angle scattering which includes momentum conservation is employed. The model includes the whole plasma inside closed flux surfaces, but the focus is on the near-axis transport due to particles with wide orbits which lead to a different heat flux than in the standard theory. Neither of two recent theories of banana regime transport near the axis is confirmed by the simulations, nor do the numerical results for the plateau regime agree with a recent theory of wide-orbit transport in this regime.

Rotation and density asymmetries in the presence of large poloidal impurity flows in the edge pedesta (invited paper)

Novel flow rotation measurements based on charge exchange recombination spectroscopy at the inboard midplane of the ASDEX Upgrade tokamak reveal the existence of an asymmetric flow structure at the H-mode edge, which is shown to arise due to a poloidal impurity density asymmetry. A quantitative evaluation of the impurity density at the inboard side demonstrates that the impurities redistribute along the flux surface, resulting in a poloidal dependency of the impurity density. The poloidal and toroidal impurity flows measured at the high-field side (HFS) and low-field side (LFS) are compared to theoretical predictions based on the parallel momentum balance, which includes friction, inertia, pressure and electric force. Both a fluid and a kinetic approach are used, showing good agreement with each other. The measured impurity flow structure is described by the model quantitatively when a finite poloidal main ion flow of ~2 km s−1 arises, which is in keeping with the standard neoclassical prediction. The interplay of all terms, in particular the inclusion of the impurity inertia term, is important in reproducing the observed flow structure and results in an impurity accumulation at the HFS. The existence of a poloidal impurity density asymmetry in the edge transport barrier slightly reduces the drift parameter v/D, however, the experimental value is consistent with standard neoclassical theory. This demonstrates that despite the asymmetry in the impurity density, the impurity particle transport is at the neoclassical level.

Collisionality dependence of the polarization current caused by a rotating magnetic island

The effect of collisions on the polarization current caused by a rotating island in a Tokamak plasma is studied with guiding center particle simulations. The simulations are performed with a δf code which includes a Monte Carlo model for Coulomb collisions. The transition between the two limiting cases of low and high collision frequency ν (compared to the island rotation frequency ω) is found to occur at ν≈ω in agreement with a recent analytic theory.

The bootstrap current in small rotating magnetic islands

The bootstrap current in small magnetic islands of neoclassical tearing modes is studied with guiding center particle simulations including pitch angle scattering. A model for a rotating island and its electric field is used and a new approximation to the electric potential in small islands is derived. Islands with sizes of the order of the ion banana orbit width are studied by means of a two-step model, which allows to treat both ions and electrons kinetically. The bootstrap current in such small islands is found to depend strongly on the direction of rotation of the island. The bootstrap current in small islands rotating in the ion diamagnetic direction is strongly diminished, similarly to what happens in big islands. In small islands rotating in the electron diamagnetic direction, on the contrary, the bootstrap current is almost completely preserved, implying a reduced neoclassical drive of the island growth.

Influence of externally applied magnetic perturbations on neoclassical tearing modes at ASDEX Upgrade

The influence of externally applied magnetic perturbations (MPs) on neoclassical tearing modes (NTM) and the plasma rotation in general is investigated at the ASDEX Upgrade tokamak (AUG). The low n resonant components of the applied field exert local torques and influence the stability of NTMs. The non-resonant components of the error field do not influence MHD modes directly but slow down the plasma rotation globally due to a neoclassical toroidal viscous torque (NTV). Both components slow down the plasma rotation, which in consequence increases the probability for the appearance of locked modes. To investigate the impact of externally applied MPs on already existing modes and the influence on the rotation profile, experimental observations are compared to modelling results. The model used here solves a coupled equation system that includes the Rutherford equation and the equation of motion, taking into account the resonant effects and the resistive wall. It is shown that the NTV torque can be neglected in this modelling. To match the experimental frequency evolution of the mode the MP field strength at the resonant surface has to be increased compared to the vacuum approximation. This leads to an overestimation of the stabilizing effect on the NTMs. The reconstruction of the entire rotation profile via the equation of motion including radial dependencies, confirms that the NTV is negligibly small and that small resonant torques at different resonant surfaces have the same effect as one large one. This modelling suggests that in the experiment resonant torques at different surfaces are acting and slowing down the plasma rotation requiring a smaller torque at the specific resonant surface of the NTM. This additionally removes the overestimated influence on the island stability, whereas the braking of the islands rotation is caused by the sum of all torques. Consequently, to describe the effect of MPs on the evolution of one island, all other islands and the corresponding torques must also be taken into account.

Kinetic calculation of the polarization current in the presence of a neoclassical tearing mode

The polarization current associated with a neoclassical tearing mode (NTM) is studied by means of drift kinetic δf simulations. This current has been invoked as a possible explanation for both the observed threshold for the minimum island size that can grow unstable and the scaling of the plasma pressure at the mode onset with the normalized gyroradius, even though the theory is not able to predict the island rotation direction and hence the role (whether stabilizing or destabilizing) of the polarization current for the island evolution. In the numerical approach presented in this paper, the island rotation frequency can be assigned as an input parameter and the corresponding behaviour of the current can be studied. The calculations are performed in toroidal geometry in the presence of a helical perturbation. It is found that kinetic effects lead to a sign reversal of the polarization current for rotation frequencies close to the diamagnetic frequency even for flat pressure profiles, thus influencing both the sign and size of the polarization-current contribution to the NTM evolution.

Monte Carlo δf simulation of the bootstrap current in the presence of a magnetic island

In the theoretical description of the neoclassical tearing mode the bootstrap current is assumed to completely vanish inside the magnetic island if finite perpendicular transport can be neglected. In this paper, the effects due to both the finite-orbit width of the trapped ions and their toroidal precession (not included in the standard analytic theory) on the island current are investigated. The evolution of the ion distribution function in toroidal geometry in the presence of a perturbed magnetic equilibrium is computed numerically employing the δf method, collisions being implemented by means of a Monte Carlo procedure. It is shown that a significant fraction of the (ion) bootstrap current survives inside the island when the ion banana width wb approaches the island width W, and no loss is observed for wb/W≥1. This effect is reduced when the collision time becomes longer than the toroidal drift time. The value of the current is found to be inconsistent with the local gradients in the island region. The finite-banana-width effect leads to a linear scaling of the value of the poloidal β at the mode onset with the normalized ion poloidal gyroradius ρ*p, in agreement with the experimental results of ASDEX Upgrade.